Can human growth hormone affect the thyroid?

No: No but if you were diagnosed with gh deficiency, it's not uncommon to have associated thyroid problems since TSH (the pituitary signal the controls thyroid production from the thyroid gland) is secreted from cells that are adjacent to the gh secreting cells.

Yes: In kids we sometimes unmask hypothyroidism once starting growth hormone when there is a gh deficiency in the first place. Once the metabolic demands of the body increase with gh use, sometimes the pituitary can't keep up its production of tsh. If the thyroid doesn't get appropriate signals to make thyroid hormone from tsh, then thyroid hormone production also goes down.