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

Participants of this study will have a diagnosis of a solid tumor cancer that has come back to its original location or spread beyond its original location (advanced), came back (relapsed) or worsened (refractory) after standard treatments, or no standard treatments are available for the participants' cancer. The purpose of this study if to find the highest dose of MQ710 that causes few or mild side effects in participants with a solid tumor cancer diagnosis.

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 \