A Study of MQ710 With and Without Pembrolizumab in People With Solid Tumor Cancer

Inclusion Criteria: * Age 18 or over * Histologically or cytologically documented advanced or metastatic cancer that has relapsed from or is refractory to standard treatment in two lines of prior therapy in the advanced setting unless there are fewer than two FDA approved lines of therapy for the particular disease, or for which no standard treatment is available * At least 2 tumors suitable for direct or ultrasound-guided injection defined as at least one cutaneous, subcutaneous, or nodal lesion or aggregate of lesions, ≥0.5 cm for any single lesion and cumulative lesion dimensions. One lesion must meet criteria for RECIST measurable disease if in Part 2. Note: One lesion will be biopsied (if possible) * Mandatory initial screening biopsy a. For patients undergoing surgical excision/resection: i. Tumor deemed accessible and safe for biopsy by the Investigator ii. Willing to consent to biopsy and surgical procedure iii. Patient able to undergo surgical procedure and appropriate anesthesia b. For patients not undergoing surgical excision/resection to obtain mandatory screening biopsy: i. Tumor deemed accessible and safe for biopsy by the Investigator ii. Willing to consent to initial tumor biopsy * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Patients with no curative treatment options available including surgery and/or definitive radiation or patients in which these modalities are associated with significant morbidity * Patients with advanced disease who have received and progressed on standard therapy or have disease for which there is no standard therapy or have contraindications to standard therapy Part 1a: Patients with cutaneous squamous cell carcinoma (cSCC), basal cell carcinoma (BCC), melanoma, Merkel cell carcinoma, sebaceous carcinoma, extramammary Paget's disease, Kaposi sarcoma, HNSCC, adnexal carcinoma, and angiosarcoma, as well as patients with cutaneous neoplasms that are separate primaries with morbidity from multiple surgeries that have failed standard therapy. Any malignancy with superficial cutaneous or subcutaneous lesions or palpable lymph nodes may be eligible based on the discretion of the investigator. * Part 2a: Patients with cutaneous squamous cell carcinoma (cSCC), basal cell carcinoma (BCC), melanoma, Merkel cell carcinoma, sebaceous carcinoma, extramammary Paget's disease, Kaposi sarcoma, HNSCC, adnexal carcinoma, and angiosarcoma, as well as patients with cutaneous neoplasms that are separate primaries with morbidity from multiple surgeries that have failed standard therapy. BCC will also be included, given that pembrolizumab has not been approved for this condition, although cemiplimab is approved. * Parts 1b and 2b: Patients must have cSCC, Merkel cell carcinoma, melanoma, or head and neck squamous cell carcinoma. These patients should be refractory to anti-PD-1 therapy, with the exception of patients with HNSCC with PD-L1 expression \<1. * Parts 1a, 2a and 2b: Patients with BRAF-mutated melanoma should have received BRAF-targeted therapy. * Predicted life expectancy of 3 months or more (in both Part 1 and Part 2) * Participant or their legally authorized representative (LAR able to provide written informed consent to participate * Ability to comply with study procedures in the Investigator's opinion * Adequate renal function as defined by Cr \<2 mg/dL * Adequate hepatic function 1. Serum bilirubin ≤1.5 x ULN 2. AST and ALT ≤2.5 ULN (no liver mets) 3. AST and ALT ≤5.0 ULN (for patients with liver mets) * Adequate bone marrow and hematologic function 1. Coagulation function adequate (PT and aPTT within x1.5 ULN) 2. Platelets ≥ 75,000/mm\^2 3. ANC ≥ 1000/uL * Females of child-bearing potential must have a negative pregnancy test within 14 days prior to enrollment and on day of treatment. All patients must agree to use adequate contraception prior to study entry, for the duration of study participation, and up to 90 days after the last dose of MQ710 * Part 2 only: at least one measurable site of disease according to RECIST criteria * Prior non-immunotherapy, anti-tumor treatment including endocrine, chemical/radiotherapy, targeted therapy, or major surgery (but not anti-PD1/- L1 therapies) was discontinued for more than 4 weeks prior to enrollment * Patients who have failed prior anti-PD1/-PDL1 may be included. Washout of anti-PD1/-PDL1 at least 3 weeks prior to initiation of therapy in Part 1a and 2a. No washout period is required for Part 1b and 2b. Exclusion Criteria: * Splenectomy * Active infections requiring antibiotics, physician monitoring or recurrent fevers (\>38.0 ℃) associated with a clinical diagnosis of active infection * Acute or chronic active viral disease or positive test for hepatitis B virus, hepatitis C virus, human immunodeficiency virus (HIV) with CD4 count \<100mg/m3, or received treatment with antivirals or nucleoside analogs such as those used in the treatment of hepatitis B (e.g. lamivudine, adefovir, tenofovir, telbivudine, entecavir), ribavirin, cidofovir, diaminopurine analogs, methyladenosine analogs, or interferon alpha within 4 weeks of initiation of study treatment .a. Patients with HIV are allowed on study if CD4 count is \>100 cells/mm3. * Incomplete recovery from surgery, incomplete healing of an incision site * Any of the following in the 3 months before the first dose of study treatment: Grade 3 or 4 gastrointestinal bleeding/haemorrhage (unless due to resected tumour), treatment-resistant peptic ulcer disease, erosive oesophagitis or gastritis, infectious or inflammatory bowel disease, diverticulitis, pulmonary embolism or other uncontrolled thrombo-embolic event, history or evidence of haemoptysis or menorrhagia * History of myocardial infarction, myocarditis, congestive heart failure (as defined by New York Heart Association Functional Classsification III or IV), unstable angina, serious uncontrolled cardiac arrhythmia,or significant cardiovascular or cerebrovascular even