The thing is, Harry's apartment has its own threshold. It might not be as secure as that of the Carpenter family's home, but it's still somewhere that a mortal has lived, has put down roots, and has cared for (I'm sure thrift store rugs count as taking care of it).
Like many of these metaphysical concepts, I would treat it more as a belief than anything physical. If this person lives in the same "place" while on the move (and you, as the GM, are okay with it), I would give him or her a threshold. Maybe not one as strong as a nice Tudor in a well-established neighborhood, but a threshold nonetheless. Ditto houseboats, if it would ever come up.
I would say that it's really up to you. It would depend on how strong of a threshold he/she wants, how they can justify it in story, and how long they'd lived in this mobile home. Your standard bachelor pad might not have much of a threshold, but anything supernatural is going to be at least aware of the fact that it's entering a home. It might not slow down anything further than a flock of pixies (do pixies flock? Swarm? School? Murder of pixies? Seems wrong to me...), but that's still a threshold.