Reported speech

Yes. I would not leave it out.

whatchadoinHe said he'd bought a jacket.Is it common to leave out 'had' in reported speech?

No, I wouldn't say so. This is all about something called "backshift", which in your example is optional.

Nevertheless, backshifting is the default option as it gives, strictly speaking, a more faithful report in that it uses the same time reference as the orig

BillJNo, I wouldn't say so.

So, you don’t think had is commonly omitted?

BillJSo, backshift preferred, but not obligatory.

The OP did not ask whether it was obligatory. I think you misread the question.

whatchadoinHe said he'd bought a jacket.Is it common to leave out 'had' in reported speech?

In this particular exanple it has not been left out. 'd is the contracted form of had.

Aspara GusBillJNo, I wouldn't say so. So, you don’t think had is commonly omitted?BillJSo, backshift preferred, but not obligatory.The OP did not ask whether it was obligatory. I think you misread the question.

I'm well aware of what the OP asked, thank you. I gave an answer with a bit of useful supporting information about backshifting.

Sorry if you p

BillJSorry if you personally don't like it.

No, I didn't dislike it. Extra information is always a good thing. I just found it odd that you wouldn't call this particular omission common, when, especially in conversation, it is.