Maybe I got it wrong, but I had thought the comment about the slippers was to let us know that the mom in this situation is so disdainful of her child's other family that she sends the kid without clothes or shoes, and to avoid sending shoes on the kid's feet, she sends the child in slippers. Did I understand that correctly?
That puts the coat thing in a different perspective -- not one of whether or not it was cold and the child needed a coat, but whether or not the *child* is the one the child support is for and if the child is entitled to her own coat, clothes, etc., during visitation. Those of us who pay child support are constantly reminded on this board that the money is for the child, not the other parent, so we shouldn't resent paying it. Fair enough, but if we paid for the kid's coat as support, shouldn't the kid get to wear it over to see us on a cold day? I mean, if the child support is only to support the kids to have a coat at mom's house, but not over here, that's a bit silly (not to mention wasteful). And if the kid's mom is so disdainful as to send the kid without clothes or outerwear, then that will not only not foster a good relationship between the homes, but it's also a misuse of child support, and, frankly, I think that's shameful.
In our situation, if the kids had no coat on a cold day, I'd be annoyed, but I wouldn't assume the worst, as I know their mom's track record (she sends clothes, at least lately and I do think this improvement will last), and so I would figure she didn't think it was as cold as we did. If she was still in the habit of not sending clothes, I'd see it completely differently.
So if I misunderstood and she does usually send the child's clothes and such, then I'd advise letting this one slide, and just buy a cheap coat just in case. Your situation is new, so hopefully it will improve. Relations between our kids' two home have improved greatly in just the past 6 months (and we've been at this for 2 years).