The horse has probably never been given a reason to have confidence in the person on its back; and so we get scenario A)
Horse sees something scary and freezes up, so rider thinks “ooooo, poor horsey it OK”, person gets all compassionate and caring for the horse, wants to show the horse that they will protect it, and take care of them, gets off the horse, leads them past the “scary thing”, gets back on, and rides to the next scary thing thinking the whole time “I really showed my horse that I will protect it. Yay”; and then they have trouble with a horse freezing up and can't figure out why or how to fix it.
The trouble is, horses don’t feel comfortable when their human starts getting all protective and stuff, what THAT tells them is that there IS something to be scared of.
What I'm guessing you will have to do with this horse is to teach it that it can have some confidence in your judgement, and to have a healthy dose of respect for you ( really, the former follows the latter). Get some good respect exercises going on the ground to get it started then carry that through to its back and you should be OK. And don’t be afraid to have a quirt or a pair of spurs ready if you need them, its better to have them and never need to use them than to need to use them and not have them. And, use them right once or twice and you shouldn’t have to use them again, they are only as tough, or harsh, as you make them.
What you will have to try to get is scenario B)
Horse freezes up when it sees something scary. Rider, who had their mind on the job at hand and was focused on where they were going/what they were doing, thinks “what the hell?” looks at horse and says “what's your problem, c'mon, lets go” and asks horse to continue like there is nothing there (which is almost always the case anyway); then makes the horse get its act together if it doesn’t keep on as it was in the first place.
Get its respect and show it, and continue to show it. That it can trust you, don’t go getting worked up about anything when it does, and it will trust you and it will keep walking out with no problems.
I guess at first you might not be able to get it going straight out, you might have to just loosen its feet up a bit, maybe get it moving sideways back and forth and/or back it up or something , to get it unstuck. But focus in your own mind on getting it moving and remain cool and the horse will pick up on that and, if it respects you, it will trust you and soon enough it wont bat an eyelid at what used to make it freeze up.
Trouble is, when you give it back to the owner, don’t be surprised if they come back to you complaining that its still doing the same old stuff, freezing and not moving.