• Participants must have histologically or cytologically confirmed respiratory papillomas that involves the lungs. Subjects can have measurable or non-measurable\* disease based on RECIST 1.1. Non-measurable disease based on RECIST 1.1 is defined as lesions with a short axis less than 10 mm.

• For those patients with non-measurable pulmonary disease, participants must have disease at other sites such as the larynx and trachea and must have undergone \> 3 surgical procedures over a 12-month period.

• Be required to provide tissue from a newly obtained biopsy of a lesion or an archived specimen. Newly-obtained is defined as a specimen obtained up to 6 weeks (42 days) prior to the first dose of study drug. Subjects for whom newly obtained samples cannot be provided (e.g. inaccessible or subject safety concern) may submit an archived specimen only upon agreement from the PI.

• Have confirmed human papillomavirus-associated lesions based on in-situ hybridization testing and/or polymerase chain reaction which may be performed on a newly obtained biopsy or archived sample.

• Age ≥18 years.

• ECOG performance status of 0 to 1.

• Participants must have adequate organ and marrow function as defined below:

• \---- Absolute neutrophil count (ANC) ≥1500/μL

• \---- Platelets ≥100 000/μL

⁃ Hemoglobin ≥9.0 g/dL or ≥5.6 mmol/L (a)

⁃ Creatinine OR Measured or calculated (b) creatinine clearance (GFR can also be used in place of creatinine or CrCl) ≤1.5 × institutional ULN OR ≥30 mL/min for participant with creatinine levels \>1.5 × institutional ULN

⁃ Total bilirubin ≤1.5 × institutional ULN OR direct bilirubin ≤ institutional ULN for participants with total bilirubin levels greater than or equal to 1.5× institutional ULN

⁃ AST (SGOT) and ALT (SGPT) ≤2.5 × ULN (≤5 × institutional ULN for participants with liver metastases)

⁃ TSH Institutional normal limit

⁃ Free T4 Institutional normal limit

⁃ Amylase less than or equal to 1.5 x institutional ULN

⁃ Lipase less than or equal to 1.5 x institutional ULN

⁃ International normalized ratio (INR) OR prothrombin time (PT)

⁃ Activated partial thromboplastin time (aPTT) ≤1.5 × institutional ULN unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants

⁃ ALT (SGPT)=alanine aminotransferase (serum glutamic pyruvic transaminase); AST (SGOT)=aspartate aminotransferase (serum glutamic oxaloacetic transaminase); GFR=glomerular filtration rate; ULN=upper limit of normal.

‣ (a) Criteria must be met without erythropoietin dependency and without packed red blood cell (pRBC)transfusion within last 2 weeks.

‣ (b) Creatinine clearance (CrCl) should be calculated per institutional standard.

∙ Note: This table includes eligibility-defining laboratory value requirements for treatment; laboratory value requirements should be adapted according to local regulations and guidelines for the administration of specific chemotherapies.

• Participants with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, participants should be class 2B or better.

• Adequately controlled blood pressure with or without antihypertensive medications defined as systolic BP ≤ 140 mmHg and diastolic BP ≤ 90 mmHg at screening with no change in antihypertensive medications within 1 week prior to screening.

• Female subject of childbearing potential must have a negative serum pregnancy test within 28 days of the first dose of study drug\*. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required.

• \*Please refer to the study calendar for requirements regarding a pregnancy test 24 hours prior to receiving any dose of study medication upon subject enrollment into the study.

• A female participant is eligible to participate if she is not pregnant or breastfeeding, and at least one of the following conditions applies:

‣ Is not a WOCBP as defined below. OR

⁃ Is a WOCBP and must be willing to use 2 methods of birth control, or abstain from heterosexual activity during the intervention period and for at least 120 days after the last dose of pembrolizumab or 30 days post lenvatinib, whichever occurs last.

• Women of childbearing potential are those who have not been surgically sterilized or have not been free from menses for greater than 1 year. The methods of surgical sterilization include having had a hysterectomy (removal of the uterus), bilateral oophorectomy (removal of both ovaries), tubal ligation (having your tubes tied), and transvaginal occlusion (blocking the tubes with a coil). The investigator should evaluate the potential for contraceptive method failure (ie, noncompliance, recently initiated) in relationship to the first dose of study intervention

• A WOCBP must have a negative highly sensitive pregnancy test (urine or serum as required by local regulations) within 24 hours before the first dose of study intervention.

• If a urine test cannot be confirmed as negative (eg, an ambiguous result), a serum pregnancy test is required. In such cases, the participant must be excluded from participation if the serum pregnancy result is positive.

• Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of pembrolizumab.

• Male participants are eligible to participate if they agree to the following during the intervention period and for at least 7 days after the last dose of Lenvatinib:

‣ Be abstinent from heterosexual intercourse as their preferred and usual lifestyle (abstinent on a long term and persistent basis) and agree to remain abstinent.

• OR

• • Must agree to use contraception unless confirmed to be azoospermic (vasectomized or secondary to medical cause) as detailed below:

• Agree to use a male condom plus partner use of an additional contraceptive method when having penile-vaginal intercourse with a WOCBP who is not currently pregnant. Note: Men with a pregnant or breastfeeding partner must agree to remain abstinent from penile-vaginal intercourse or use a male condom during each episode of penile-vaginal penetration.

• Contraceptive use by men should be consistent with local regulations regarding the methods of contraception for those participating in clinical studies. If the contraception requirements in the local label for any of the study interventions is more stringent than the requirements above, the local label requirements are to be followed

‣ Ability to complete Patient Medication and Blood Pressure diaries by themselves or with assistance.

⁃ Ability to understand and the willingness to sign a written informed consent document.