Docetaxel and SX-682 in Recurrent/Metastatic Head and Neck Squamous Cell Carcinoma, Salivary Gland Carcinoma, and Advanced Prostate Cancer

* INCLUSION CRITERIA:

All Participants

* Age \>= 18 years.
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) \<= 2
* Participants must have adequate organ and marrow function as defined below:

  * ANC \>= 1,500/mcL
  * Hemoglobin (Hgb) \>= 9 g/dL
  * Platelets (PLTs) \>= 100,000/mcL
  * Creatinine clearance \>= 50 mL/min (by Cockroft-Gault formula)
  * Total bilirubin \<= 1.5 x iULN (\<= 3 x ULN in participants with known/suspected Gilbert s disease)
  * ALT/AST \<= 2.5 x iULN
  * Activated partial thromboplastin time (aPTT) \<= 1.5 x iULN
* Contraception as follows:
* Women of child-bearing potential (WOCBP) must agree to use an effective method of contraception (barrier, hormonal, intrauterine device \[IUD\], surgical sterilization, abstinence) prior to study entry, for the duration of study treatment, and for up to 2 months after discontinuation of the study drugs. A participant may request a male partner to use an effective form of contraception to fulfill this requirement.
* Men able to father a child must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) for the duration of the study treatment and up to 4 months after discontinuation of the study drugs. A participant may request a female partner to use an effective form of contraception to fulfill this requirement. Men able to father a child must not freeze or donate sperm within the same period.
* Nursing participants must be willing to discontinue nursing from study treatment initiation through one week after the last dose of study drugs.
* Participants must be able to swallow oral medications.
* Human immunodeficiency virus (HIV)-infected participants must have undetectable viral load (VL) and be on effective anti-retroviral therapy within 4 weeks prior to the study treatment initiation and have no history of opportunistic infections or Castleman s disease within 12 months prior to the study treatment initiation.
* Participants with evidence of chronic hepatitis B virus (HBV) infection must have undetectable HBV VL.
* Participants with evidence of chronic hepatitis C virus (HCV) infection must have undetectable HCV VL.
* Participants must be able to understand and willing to sign a written informed consent document.

Participants with HNC

* Histologically confirmed HNSCC (including oral cavity, oropharynx, larynx, hypopharynx, paranasal sinuses, nasopharynx) or SGC (including ACC and non-ACC) and recurrent/metastatic (R/M) or advanced incurable disease.
* Prior treatment as follows:

  * Participants with R/M HNSCC must have prior systemic treatment (platinum-based chemotherapy and/or anti-PD(L)1 treatment).
  * Participants with R/M SGC may have any number of prior systemic treatment lines; prior systemic treatment not required for participation.
  * Participants must not have received systemic anticancer treatment within 3 weeks prior to first treatment administration. Note: Treatment-related toxicities must have resolved to Grade \<2 or be minimal and not constitute a safety risk. Participants with SGC previously treated with hormonal therapies (e.g., drugs targeting the androgen receptor) may continue these drugs concomitantly with study therapy. Participants with bone metastases or hypercalcemia on intravenous bisphosphonate medications, denosumab, or similar agents, are eligible to participate and may continue this treatment.
* Presence of \>= 1 measurable lesion by RECIST v 1.1 criteria.

Participants with mCRPC

* Documented histopathological confirmation of prostate cancer. If no pathologic report or specimen is available, participants may enroll with a history of clinical course consistent with the disease.
* Participants must have mCRPC, defined as at least one lesion on TC-99 bone scan or at least one lesion that is measurable per RECIST 1.1.
* Participants must need ADT as part of their cancer therapy (unless previous orchiectomy)
* Castrate testosterone level (\<50 ng/dl or 1.7 nmol/L)
* Prior treatment as follows:

  * DTX for mCRPC is allowed but participants must not have had progression while on docetaxel or within 3 months after completing DTX for mCRPC
  * Participants must have been previously treated with modern anti-androgens such as abiraterone, enzalutamide, apalutamide, or darolutamide.
* Progression defined as two consecutive rising PSA values at least 1 week apart or radiographic evidence of progression seen on computed tomography (CT) scan or TC- 99 bone scan.
* Toxicities related to prior therapy, including surgery and/or radiation, must have resolved to \< Grade 1 per CTCAE v.6.0.

EXCLUSION CRITERIA:

All participants

* History of allergic reactions attributed to compounds of similar chemical or biologic composition to DTX, SX-682, or other agents used in study (e.g., polysorbate 80).
* Known active brain metastases. Note: Participants with previously treated brain metastases are eligible if imaging at least four weeks prior to first trial treatment shows no evidence of progression and neurologic symptoms have resolved, have no new or enlarging brain metastases, and are not using glucocorticoids for at least a week prior to first trial treatment
* Participants must not have received other investigational agents within 3 weeks prior to the first dose of the study drug(s).
* Participants must not have received major surgery within 14 days prior to the first drug administration (minimally invasive procedures such as diagnostic biopsies are permitted). If participant underwent major surgery, they must have recovered adequately (according to the Principal Investigator) from the toxicity and/or complications from the intervention prior to starting study treatment.
* Treatment (systemic) with any medications or substances that are moderate or strong inducers or moderate or strong inhibitors of cytochrome P450 (CYP3A4) listed at https://www.fda.gov/drugs/drug-interactions-labeling/drug-development-and-druginteractions- table-substrates-inhibitors-and-inducers#table2-2,table3-3,table5-2 within 2 weeks or 5 half-lives (whichever is shorter) prior to the start of the study treatment.
* Prior or concurrent malignancy whose natural history or treatment has potential to interfere with the safety or efficacy assessment of the study treatment.
* Participants with serious uncontrolled intercurrent illness evaluated by medical history, electrocardiogram (EKG), and physical exam that would unacceptably increase risk for the participant or impair the ability to evaluate the endpoints of the study or that would limit compliance with study requirements.

Participants with HNC

* Participants must not have received large-field radiotherapy within 2 weeks prior to first treatment administration. All residual treatment-related toxicities must have resolved to Grade \<2 (except for radiation-induced xerostomia/dysgeusia) or be minimal and not constitute a safety risk.
* Positive pregnancy serum or urine beta-human chorionic gonadotropin (beta-hCG) test

Participants with mCRPC

* Use of other medications for urinary symptoms including 5-alpha reductase inhibitors (finasteride and dutasteride) and alternative medications known to alter PSA (e.g., phytoestrogens and saw palmetto) within 1 week prior to the study treatment initiation.
* Cancer related neuropathy at screening
* Baseline QTcF \>= 470 ms