Does the oath depend on the intention of the swearer or the one who administers the oath, if a man swears to his wife that he did not do something – when he actually did it – and then secretly makes an exception to himself, “except for one time”? And what is the expiation for this oath if it was a false oath?

If the oath is not for a legal adjuration, but to reassure the woman and remove her doubt, then what matters is your intention and your reservation in it. Your specification of it is acceptable as long as the wording allows for it. It is stated in the Fiqh Encyclopedia that the intention of the one taking the oath is considered if there was no adjuration at all, or if it existed but lacked one of its conditions. In Islamic jurisprudence and its evidence, a non-judicial oath is based on the intention of the one taking it, and he is permitted to use equivocation (tawriyah) in it.