Disclaiming Banner: If someone sues me for not owning ATLA they will win hysterical laughter at the trial.
Piggy-back
Her feet were burned pretty badly. And it was his fault so Zuko just didn't bother to complain when she insisted on being carried that first week. Even if Toph wiped her nose on his shirt or yanked on his hair.
It was probably a good thing in some respects. Katara was a little less grouchy when the tiny terror was on time for meals. And she didn't fight it as much when she didn't realize that they were heading to the bathing rooms. It went a long ways toward Aang really trusting him as a teacher, too.
But Toph herself...Zuko remained a little bit amazed all that time Toph insisted that he carry her around. It had been years since anyone had trusted him.