Inclusion Criteria:

* Age ≥ 18 years of age
* Have an Eastern Cooperative Oncology Group (ECOG) performance status of ≤ 2
* RRMM patients with one or more prior lines of therapy with at least one ES or PS lesion that is accessible to a biopsy. Accessibility will be assessed by the MM tumor board
* Measurable disease meeting at least one of the following:

* Serum M-protein ≥1 g/dL
* Urine M-protein ≥200 mg/24 h
* Serum FLC assay: involved FLC level ≥10 mg/dL provided serum FLC ratio is abnormal
* Up to 10 patients without measurable disease can be enrolled but screening imaging and/or bone marrow biopsy have to confirm RRMM. Follow-up response assessment will be performed with imaging using RECIST 1.1 and Deauville Criteria and bone marrow biopsies
* Absolute neutrophil count: ≥ 1 x 10\^9/L
* Platelets: ≥ 75 x 10\^9/L
* Total bilirubin: ≤ 1.5 x upper limit of normal (ULN)
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]): ≤ 3 x ULN
* Renal function: Estimated creatinine clearance ≥ 30 mL/min (Cockroft-Gault)
* Adequate cardiac pump function with a left ventricular ejection fraction of ≥ 40%
* Women of child-bearing potential must agree to use adequate contraceptive methods (e.g., hormonal or barrier method of birth control; abstinence) prior to study entry and for at least 28 days after the last dose of mezigdomide or 6 months after the last dose of carfilzomib. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately
* Male patients (non-vasectomized) must agree to use contraception during the treatment period and for at least 28 days after the last dose of mezigdomide or 3 months after the last dose of carfilzomib and refrain from donating sperm during this period
* Participant must understand the investigational nature of this study and sign an independent ethics committee/institutional review board approved written informed consent form prior to receiving any study related procedure

Exclusion Criteria:

* Participant has a history of anaphylaxis or hypersensitivity to thalidomide, lenalidomide, pomalidomide (including ≥ grade 3 rash during prior thalidomide, lenalidomide, or pomalidomide therapy), carfilzomib or dexamethasone, any cereblon E3 ligase modulators (CELMoD) agents, or the excipients contained in the formulations, or participant has any contraindications per local prescribing information
* Administration of strong CYP3A modulators or proton-pump inhibitors (e.g., omeprazole, esomeprazole, lansoprazole, pantoprazole, rabeprazole) within 2 weeks of starting study intervention
* Participant is unable or unwilling to undergo protocol required thromboembolism prophylaxis
* Patient has evidence of mucosal or internal bleeding and/or is platelet transfusion refractory
* Any medical conditions that, in the investigator's opinion, would impose excessive risk to the patient or would adversely affect his/her participation in this study
* Known active infection requiring parenteral or oral anti-infective treatment within the past 14 days
* Participant has a history of prior malignancy other than MM, except if the participant has been free of disease for ≥ 3 years or the participant had 1 of the following noninvasive malignancies treated with curative intent without known recurrence:

* Basal or squamous cell carcinoma of the skin
* Carcinoma in situ of the cervix or breast
* Stage 1 bladder cancer
* Incidental histological findings of localized prostate cancer such as tumor stage 1a or 1b (T1a or T1b) using the tumor, nodes, and metastasis (TNM) classification of malignant tumors OR prostate cancer that has been treated with curative intent
* Other ongoing anti-myeloma therapy. Patients may be receiving concomitant therapy with bisphosphonates and low dose corticosteroids for symptom management and comorbid conditions. Doses of corticosteroid should be stable for at least 7 days prior to patient registration
* Pregnant or breast-feeding females
* Serious psychiatric illness, active alcoholism, or drug addiction that may hinder or confuse follow-up evaluation
* Known active HIV or hepatitis B or C viral infection
* Known history of HIV infection
* Systemic amyloidosis or POEMS syndrome (plasma cell dyscrasia with polyneuropathy, organomegaly, endocrinopathy, monoclonal protein \[M-protein\] and skin changes)
* Prior peripheral stem cell transplant within 12 weeks of study enrollment
* Radiotherapy within 14 days prior to cycle 1 day 1. However, if the radiation portal covered ≤ 5% of the bone marrow reserve, the patient may be enrolled irrespective of the end date of radiotherapy
* Known intolerance to steroid therapy
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, severe cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Carfilzomib-refractory in the most recent line of therapy
* Prior treatment with mezigdomide
* Contraindication against conscious sedation
* Unwilling or unable to follow protocol requirements
* Any condition which in the investigator's opinion deems the participant an unsuitable candidate to receive study drug