Is

It was me that Mr. Jones thought could do it.

correct?

Is that is required between thought and could?

Is

It was me that Mr. Jones thought could do it.

correct?

Is that required between thought and could?

Is

It was me that Mr. Jones thought could do it.

correct?

Or is there a that required between could and do?

EDIT: Mistake on my part. What I meant was if "that" is required between "thought" and "could". Sorry for the confusion.

Is

It was me that Mr. Jones thought could do it.

correct?

Is that is required between thought and could?

Is

It was me that Mr. Jones thought could do it.

correct?

Or is there a that required between could and do?

Is

It was me that Mr. Jones thought could do it.

correct?

Or is there a that required between could and do?

EDIT: Mistake on my part. What I meant was if "that" is required between "thought" and "could". Sorry for the confusion.