It sounds highly unlikely. But, it depends what you mean by a 'poor diet'. Maybe you're eating something that causes an allergic reaction. Perhaps you should note what you're eating and see if there is anything that stands out as always being present when you have a sore throat - if you see what I mean.
As for feeling sluggish after a few days eating badly, if you don't get enough protein, or too much sugar, then yes, it's quite possible. But, I don't know that that would have an effect on your thyroid.
If you feel as though you have a lump in your throat as you swallow you should ask your GP to palpate your throat and neck to feel for a goitre or nodule. If such is detected you should be referred for an ultrasound scan.
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