Inclusion Criteria:  Understands and provides written informed consent and willing to follow the requirements specified in protocol. Age ≥18 years. Life expectancy ≥6 months. For Part B patients, documented Tissue Factor (TF) presence in tumor biopsy specimens obtained from archival or re-biopsy specimens by immunohistochemistry (IHC) protein expression. Must have histologically or cytologically confirmed unresectable or metastatic cancer with documented disease progression during prior therapy, or relapse or progression following approved standard therapy for their tumor types- Part A and Part B. Part B: Patients who have documented progression during or relapse following standard therapy, no further treatment options that are known to improve survival, and participation in a clinical trial is a reasonable therapeutic option. Patients must have measurable disease per RECIST v1.1. ECOG performance status of 0 or 1. Acceptable bone marrow, hepatic, cardiac, renal, and coagulation function. A negative serum pregnancy test if female and aged between 18-55 years old. Patients, both females and males, of reproductive potential must agree to use adequate contraception during and for 180 days after the last infusion of MRG004A.  Exclusion Criteria:  Archival or biopsy tumor shows TF IHC membrane or cytosolic score of zero, no TF-positive expression or no TF-positive staining in Part B patients. Toxicities (except alopecia & fatigue) due to prior antitumor therapy are greater than CTCAE v5.0 Grade 1. Toxicities due to prior radiotherapy that have not resolved to Grade ≤ 1 CTCAE v5.0 at least 21 days prior to the first treatment. Untreated, unstable or uncontrolled central nervous system (CNS) metastases. Any other type of anti-cancer therapy within 21 days of the first dose of study treatment. Use of any other type of anti-cancer treatment is prohibited throughout the study. Patients with increased bleeding risk. Presence of severe cardiac dysfunction. Pulmonary embolism or deep vein thrombosis within 3 months prior to the first dose of study drug. Concurrent malignancy within 5 years prior to entry. Uncontrolled or poorly controlled hypertension. History of ventricular tachycardia, or torsade des pointes. History of moderate to severe dyspnea at rest. Major surgery within 4 weeks of the first dose of study treatment and not fully recovered. Minor surgery within 2 weeks prior to study treatment. Known allergic reactions to any component or excipient of MRG004A or known allergic reactions to other prior anti-TF (including investigational) or other monoclonal antibody ≥ Grade 3. Patients who have any known liver disease, including chronic hepatitis B, hepatitis C, autoimmune hepatic disorders, primary biliary cirrhosis or sclerosing cholangitis; Patients who have concurrent, serious, uncontrolled infections or known infection with HIV, or have a diagnosed acquired immunodeficiency syndrome (AIDS); or an uncontrolled autoimmune disease, or have undergone organ transplant. Active uncontrolled bacterial, viral, fungal, rickettsial, or parasitic infection. Use of systemic corticosteroids within 4 weeks prior to the first dose of treatment. Use of strong CYP3A4 inhibitors or inducers with MRG004A. Other excluded medications or treatment: therapeutic anti-coagulative, or long-term anti-platelet treatment; multivitamins, calcium, vitamin D, and prophylactic anti-RANKL (denosumab) and zoledronic acid therapies for bone metastases are allowed. Any patient with a positive pregnancy or is breast-feeding. Any severe and/or uncontrolled systemic disease that at the discretion of investigator and sponsor makes it undesirable for the patient to participate in this study.