Inclusion Criteria:  Metastatic or locally advanced solid tumor malignancies (breast cancer, urothelial cancer, ovarian cancer, melanoma, NSCLC, renal cell carcinoma, squamous cell cancer of the head and neck, colorectal cancer, and hepatocellular carcinoma) that has progressed on, is refractory to, intolerant to, or for which there is no curative standard of care therapy. Measurable disease with at least 1 lesion amenable to response assessment per RECIST 1.1. Demonstrate adequate organ function. All screening laboratories should be performed within 14 days of treatment initiation. Has a performance status of 0-2 on the ECOG Performance Scale. Life expectancy >12 weeks at baseline.  Women of childbearing potential must have negative serum or urine pregnancy test within 72 hours prior to receiving the first study drug administration. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required.  Women of childbearing potential must be willing to use an adequate method of contraception from 30 days prior to the first study drug administration and 120 days following last day study drug administration.  Male patients of childbearing potential must be surgically sterile, or must agree to use adequate method of contraception during the study and at least 120 days following the last day of study drug administration. Age ≥18 years at screening. Able and willing to provide written informed consent and to follow study instructions.  Exclusion Criteria:  Patient has disease that is suitable for therapy administered with curative intent. Untreated or uncontrolled central nervous system (CNS) involvement. Any concurrent or recent use (within 28 days or 5 half-lives, whichever is longer) of chemotherapy, immunotherapy, or biologic or hormonal therapy for cancer treatment. Unresolved toxicities from prior therapy, defined as having not resolved to NCI CTCAE v5.0 Grade 0 or 1, with exception of alopecia and vitiligo. Systemic corticosteroids at doses exceeding 10 mg/day prednisone or equivalent. Patient has an active infection requiring systemic therapy. Patients who have known active HIV, Hepatitis or active COVID-19 infection. (Patients who have been vaccinated against Hepatitis B and who are positive only for the Hepatitis B surface antibody are permitted to participate in the study). Patients who are positive for hepatitis B or C virus must be tested for and have an undetectable viral load.  Patients with unstable/inadequate cardiac function:  New York Heart Association Class 3 or 4 congestive heart failure, Uncontrolled hypertension, Acute coronary syndrome within 6 months, Clinical important cardiac arrhythmia, A marked baseline prolongation of QT/QTc interval (e.g., repeated demonstration of a QTc interval >470 milliseconds (ms) (CTCAE grade 1) using Fredericia's QT correction formula. A history of additional risk factors for Torsades de Pointes(TdP) (e.g., heart failure, hypokalemia, family history of Long QT Syndrome). The use of concomitant medications that prolong the QT/QTc interval. Concomitant use or recent use (at least 2 weeks before the start of ZX-4081 treatment) of drugs that are known to be strong CYP3A4/5 inhibitors and CYP3A4/5 inducers (See 5.4.2 for a list of study medications not allowed during this study). Concomitant use of drugs that are sensitive substrates with narrow therapeutic index for one of the following: CYP2C9, CYP2C19, CYP2C8, CYP2D6 and CYP3A4 (See 5.4.2 for a list of study medications not allowed during this study). Concomitant use of drugs that are sensitive substrates with narrow therapeutic index for one of the following transporters: P-glycoprotein (P-gp) and Breast Cancer resistance Protein (BCRP). Recent use a proton pump inhibitor (within 4 days prior to the start ZX-4081 treatment) or a histamine H2 blocker (within 2 days prior to the start of ZX-4081 treatment). Uncontrolled concurrent illness. Patient has concurrent active malignancy other than nonmelanoma skin cancer, carcinoma in situ of the cervix, or superficial bladder cancer who has undergone potentially curative therapy with no evidence of disease. Patients with other previous malignancies are eligible if disease-free for >2 years. Participation in another clinical trial of an investigational agent within 30 days of screening. Patient has known psychiatric, substance abuse or other disorders that would interfere with cooperation with the requirements of the trial, in the opinion of the investigator. Patients who are pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the screening visit through 30 days after the last dose of trial treatment.