May/might had done

Question

We can use the structure “may/might have done” to talk about what was possible to happen or the past possibilities:

You might have left your cellphone at work. = Perhaps you have left your cellphone at work.

I wonder if we can use a similar structure for the past. For example:

I could not find my cellphone. I might had left it at work. = Perhaps I had left it at work.

I have just seen the first structure in my grammar textbook but not the second one. Can we use the structure “might had done”, like in the second example?

Answer 1

The structure 'might had' is ungrammatical. If you are using a modal auxiliary verb (might, may, could, etc) to express something in the past, it has to be in the format have + past participle.

I might have passed the exam if I had studied a harder.
You might have said something that made him angry.
He may have been at the party, but I can't be sure.

Answer 2

may and might take the bare infinitive of the verb that follows

I may [to] go ---> I may go

I might [to] go ---> I might go

I may [to] have Covid 19 ---> I may have Covid 19

I might [to] have Covid 19 ---> I might have Covid 19

I may [to] have gone ---> I may have gone

I might [to] have gone ---> I might have gone

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No English verbs have a simple-past infinitive

You can say "to say". You cannot say "to had".

I may [to] had gone ---> I may had gone.

I might [to] had gone ---> I might had gone