Is there a cure for underactive thyroid?

No: Of course, much depends on the underlying cause , but generally, there is no "cure." there is treatment, in the form of supplemental thyroid hormone, called levothyroxine.

Typically, no.: Once the thyroid makes too little thyroid hormone it typically does not regain normal function on its own. Under active thyroid is treated by taking thyroid hormone as a medication so that the amount in the blood becomes normal.

Although: There are some conditions that cause transient (temporary) low functioning thyroid, and those may reverse on their own (will cure itself)- such sick euthyroid or some phases of thyroiditis. Or rarely in the US, iodine deficiency associated hypothyroidism may be completely reversed with eating iodine supplemented food.