Hello and welcome to HCM.
The short answer to your question is: No, you should not take supplemental Iodine to restore your thyroid gland function which was partially reduced by
radioactive iodine therapy.
Your regular diet contains enough iodine for your needs. Iodine is present in table salt, Coarse-grained sea salt, and seafood. (Kosher salt does not have iodine)
You should know that it is understood that taking too much iodine may actually result in subclinical
hypothyroidism. It's a paradoxical situation since hypothyroidism is sometimes a result of
iodine deficiency (at least it used to be in the past when salt was not fortified with iodine). These days iodine defficiency is very rare.
It was shown that relatively higher doses of supplemental iodine — 400 micrograms a day caused development of subclinical hypothyroidism.
In your case where your thyroid gland was partially destroyed by radioactive iodine, your hypothyroidism is caused not by lack of iodine but by loss of thyroid gland tissue which is capable of producing thyroid hormone.
The treatment for this is replacement therapy with thyroid hormones.
Synthroid is the most commonly used one.
The symptoms which you described: "no energy
constipation muscle pain and hair loss" are not symptoms/side effects of synthroid but of untreated hypothyroidism.
You should have your
thyroid function tests ( TFTs) re-checked and discuss with your doctor the best medication to replace your thyroid hormones.
I hope you find my answer helpful.
Regards and wishing you best of health.