Second thoughts are fresh doubts about a decision you already made. Have second thoughts, on second thoughts, no second thoughts at all.
On second thoughts is the fixed phrase used to change your mind mid-sentence. Americans usually drop the s and say on second thought.
1phrasesecond thoughtsFresh doubts about a decision you already made.
“She was having second thoughts by the time the papers arrived.”
second thoughts is a phrase.
Also accepted in games: have second thoughts