Harding had at least one mistress and at least one illegitimate child, but I doubt girlfriends traveled with him and his wife.
He wasn't Jack Kennedy for Pete's sake.
Ma, ma, where’s my pa?
Off to the White House, ha ha ha!
(That was a few decades before Harding.)
It was Washington who said we should not get entangled with foreign affairs, but he never said anything about domestic.