People with HIV experience earlier impairments in physical function compared to people in the general population. They also exhibit an earlier presentation and more rapid development of frailty, a multisystemic syndrome of aging characterized by reduced activity, fatigue, slowness, weakness, and weight loss. While exercise can improve physical function in people with HIV, it is less effective in doing so than in the general population and is difficult to sustain in the long-term. The goal of this clinical trial is to learn whether the medication tesamorelin will improve physical function and muscle health in adults with HIV when combined with exercise. Tesamorelin is a growth hormone-releasing hormone analogue that is FDA-approved to treat abdominal fat accumulation in people with HIV. While tesamorelin has also been shown to increase muscle mass and improve measures of muscle health, its effects on physical performance and muscle strength have not yet been evaluated. During a 24-week intervention phase, half of participants will be randomly assigned to receive tesamorelin and half of participants will be randomly assigned to receive placebo (a look-alike substance that contains no drug). All participants also will engage in a home-based exercise intervention supervised by an exercise coach. During a subsequent 24-week extension phase, individuals will be monitored off study drug and supervised exercise, and be encouraged to continue to exercise independently. The investigators will investigate effects of tesamorelin on physical function, muscle mass and quality, quality of life, and exercise adherence and self-efficacy. They also will evaluate whether effects of tesamorelin are maintained following treatment cessation. This study may identify an important strategy to improve how individuals aging with HIV function and feel with potential applications to other patient populations.