Inclusion Criteria

Patients must have a diagnosis of recurrent, persistent, or progressive myelodysplastic syndrome (MDS) or chronic myelomonocytic leukemia (CMML) according to WHO 2016 diagnostic criteria:

* For patients with higher-risk MDS (Intermediate, High, or Very High by IPSS-R, score \>3.5), they must have been unresponsive to 4 cycles of decitabine or 6 cycles of azacitidine, progressed on any prior HMA, or intolerant of prior treatment with either azacitidine or decitabine (HMA) chemotherapy.
* For patients with CMML, they must have received prior HMA chemotherapy and have been unresponsive to, progressed on, or be intolerant of this therapy; or those who decline or are not felt to be candidates for HMA chemotherapy.
* Patients with MDS or CMML that has recurred after prior allogeneic stem cell transplantation.
* For patients with lower-risk MDS (Low or Intermediate by IPSS-R, score ≤ 3.5, or any score post-HMA), they must be transfusion-dependent according to IWG criteria and must either be unresponsive to/progressed after prior ESA therapy or have an erythropoietin level \> 500 U/L, who are not felt to be candidates for or lack other treatment options.
* Female and male patients aged ≥ 18 years.
* ECOG performance status ≤2.
* All participants must have adequate organ and marrow function as defined below within 21 days prior to study enrollment:
* Total bilirubin \< 1.5 mg/dL unless due to Gilbert's (\<3.0 mg/dL)
* AST(SGOT)/ALT(SGPT) ≤ 2.5 × institutional upper limit of normal
* INR \< 1.5x ULN
* Creatinine clearance (estimated GFR) ≥45 mL/min/1.73 m2 as measured by the Cockroft-Gault formulation
* Patients enrolled on Part 1 (Dose Finding) of the study need not have splicing factor mutation status known at the time of enrollment but local testing must be initiated or a sample appropriately stored for future testing prior to receiving drug.
* Patients enrolled during Part 2 (Dose Expansion) of the study must have the results of splicing factor mutation testing, per local testing practices, prior to enrollment. Assessment can be performed at any time at or after MDS/CMML diagnosis.
* Part 2 patients enrolled on the splicing factor mutated arm must have at least one of the following mutations:

SF3B1: E622 Y623 R625 N626 H662 T663 K666 K700 V701 I704 G740 K741 G742 A774 D781 U2AF1: S34 R156 Q157 SRSF2: P95 Deletion including amino acid P95 ZRSR2 Any frameshift or nonsense mutation

* Part 2 patients enrolled on the splicing factor wildtype arm should lack the above mutations according to local testing. If patients have an alteration in a splicing factor that is not listed they would be included in the wildtype arm.
* Eligible patient are not currently considered candidates for, or have declined, stem cell transplantation at the time of enrollment.
* The effects of AZD6738 on the developing human fetus are unknown. For this reason and because agents that inhibit ATR may be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 6 months after the last dose of a study drug.

* Females must not be breast feeding and must have a negative pregnancy test prior to start of dosing if of child-bearing potential or must have evidence of non-child-bearing potential by fulfilling one of the following criteria at screening:

* Post-menopausal defined as aged more than 50 years and amenorrhoeic for at least 12 months following cessation of all exogenous hormonal treatments.
* Documentation of irreversible surgical sterilisation by hysterectomy, bilateral oophorectomy or bilateral salpingectomy but not tubal ligation.
* Amenorrhoeic for 12 months and serum follicle-stimulating hormone (FSH), luteinizing hormone (LH) and plasma oestradiol levels in the postmenopausal range for the institution.
* Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and for 1 week after the last study drug administration. Sexually active male patients must be willing to use barrier contraception i.e., condoms with all sexual partners. Where the sexual partner is a 'woman of child-bearing potential' who is not using effective contraception, men must use a condom (with spermicide) during the study and for 6 months after the last dose of a study drug.
* Ability to swallow and retain oral medication.
* Ability to understand and the willingness to sign a written informed consent document. Provision of informed consent should occur prior to any study-specific procedures.

Exclusion Criteria

* Involvement in the planning and/or conduct of the study (applies to both Investigator staff and/or staff at the study site).
* Previous enrollment in the present study.
* Participants with a diagnosis of ataxia telangiectasia.
* Participants who have had systemic chemotherapy within 21 days or five half lives of the medications used, whichever is longer, prior to entering the study.

• Hydroxyurea is allowed if necessary for count control.
* Patients who have received treatment with a small molecule investigational medicinal product (IMP) within 21 days or five half-lives (or 42 days for biologics), whichever is longer, prior to first dose of study drug.
* Participants who have undergone major surgery within 28 days before first dose of study drug.
* Participants who, with the exception of alopecia, have not recovered from any adverse events ≥ Common Terminology Criteria for Adverse Events (CTCAE) grade 2, unless those toxicities are deemed irreversible and unlikely to interfere with participation on the study per the treating investigator.
* Participants who have had a prior allogeneic transplant must be at least 100 days out from transplant "day 0" and off systemic immunosuppressive therapy without active grade \> 1 GVHD requiring more than 10mg prednisone/day or equivalent.
* Patients receiving systemic corticosteroids may not be on a dose of \> 10mg prednisone/day or equivalent up to 14 days prior to first dose of study drug.
* Participants who are currently receiving any other investigational agents.
* Participants with a diagnosis of/progression to acute myeloid leukemia, per WHO 2016 diagnostic criteria.
* Active CNS involvement of leukemia; evaluation (e.g. lumbar puncture) is not necessary in absence of clinical suspicion.
* Participants with a known hypersensitivity to AZD6738 or any excipient of the product.
* Hematuria +++ on microscopy or dipstick.
* Participants with prior exposure to an ATR or other DDR inhibitor.
* Participants may not be receiving any medications or substances that are potent inhibitors or inducers of CYP3A4.

* There is a required wash-out period of 5 half-lives from such agents prior to starting AZD6738, or three weeks for St. John's Wort.
* For non-potent inhibitors or inducers of CYP3A4, the decision to allow a patient to enroll on the study will be made on a case-by-case basis with the Investigator and the Overall PI. Note these include common azole antifungals, macrolide antibiotics, and other medications listed in the concomitant medications section. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product.
* Exposure of other drugs metabolised by CYP3A4 and/or CYP2B6 may be reduced and additional monitoring may be required (see Appendix G).
* The use of herbal supplements or 'folk remedies' (and medications and foods that significantly modulate CYP3A activity) should be discouraged. If deemed necessary, such products may be administered with caution and the reason for use documented in the CRF (see Appendix G).
* Uncontrolled intercurrent illness including, but not limited to, serious and active uncontrolled infection, symptomatic congestive heart failure, active inflammatory bowel disease, unstable respiratory or cardiac conditions, unstable angina pectoris, unstable cardiac arrhythmia, active bleeding diathesis (e.g. haemophilia or von Willebrand disease), uncontrolled seizures, or psychiatric illness/social situations that would limit compliance with study requirements.
* Any of the following cardiac diseases currently or within the last 6 months:

* Unstable angina pectoris or acute myocardial infarction.
* Congestive heart failure (NYHA ≥ Class 2) or known reduced LVEF \< 50%.
* Unstable conduction abnormality not controlled with pacemaker or medication e.g. third-degree heart block.
* Unstable ventricular or supraventricular arrhythmias (patients with chronic controlled atrial arrhythmias in the absence of other cardiac abnormalities are eligible).
* Symptomatic carotid stenosis, TIA, haemorrhagic or thrombotic stroke.
* Cardiac procedures such as CABG, angioplasty or vascular stenting within 6 months of dosing.
* Patients with relative hypotension (\<90/60 mmHg) or clinically relevant orthostatic hypotension, including fall in blood pressure of \> 20 mm Hg.
* Resting corrected QT interval (QTc) \>470 msec for females and \>450 for males (Fredericia formula). Patients with known factors that increase the risk of QTc prolongation such as a personal/immediate family history of long QT syndrome are ineligible.
* Active untreated or concurrent malignancy that is distinct in primary site or histology, excluding: non-melanoma skin cancer, noninvasive colonic polyps, superficial bladder tumors, cervical cancer in-situ, ductal carcinoma in situ of the breast, monoclonal B-cell lymphocytosis, or monoclonal gammopathy of undetermined significance. Patients may be receiving maintenance hormonal therapy for prior breast or prostate cancer. Other malignancies that were treated with curative intent at least 3 years prior to study screening and without evidence of active disease will be allowed.
* Female patients who are pregnant, breast-feeding or male or female patients of reproductive potential who are not employing an effective method of birth control. Pregnant women are excluded from this study because AZD6738 is chemotherapy agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with AZD6738, breastfeeding should be discontinued if the mother is treated.
* HIV-positive participants, including those on combination antiretroviral therapy, are ineligible, both because of the potential for pharmacokinetic interactions with AZD6738, as well as an increased risk of lethal infections when treated with marrow-suppressive therapy. Screening for patients without suspected disease is not required. Appropriate studies will be undertaken in participants receiving combination antiretroviral therapy when indicated.
* Patients who are known to have active infection with hepatitis B or hepatitis C based on serologic status. Screening for patients without suspected disease is not required. Patients treated with viral suppression are eligible.
* Patients who have been vaccinated with live, attenuated vaccines within 4 weeks of first dose of study drug.
* Refractory nausea and vomiting, chronic gastrointestinal diseases or previous significant bowel resection, with clinically significant sequelae that would preclude adequate absorption of AZD6738