This sounds like the story First Flight from Fire by Robin McKinley and Peter Dickinson.
The description of the story is:
"First Flight" follows the story of a younger brother, long lost in the shadows of his older siblings, who wants to pursue the disreputable profession of a healer. But when his dragon-rider brother's first flight is assigned to a dragon who has lost an eye and can no longer travel through fire space, he goes to the dragon-riding academy, even though everyone knows there is nothing that anyone could possibly do.
The protagonist's pet is Sippy who is a type of miniature dragon called a foogit:
Mum and Dad and I were at a craft fair once and a foogit pup got
trod on by a horse. Nobody knew who the pup belonged to and it
was lying there crying with its leg at a funny angle and all sort of
mashed looking.
I splinted the pup’s leg as best I could from having watched Ralas do
it a lot better, although it was harder than that because, as I say, it
wasn’t just broken, it was kind of crunched up. There were a couple
of places where the bone poked through the skin and I didn’t have
a clue what to do about that, I just made a big green smelly poultice
of pretty much any plant I could find that hadn’t been stomped fl at
by everybody at the fair and slapped it on. Mum was really annoyed,
but mostly about my shirt (she said something about vermin too but I think she meant fleas) but she let me keep him till we got
home. Nobody came around asking about him either.
But we didn’t get home till ten days later and the bones had already started to knit (Ralas said) and, as I say, I didn’t know what I
was doing so the bones healed a bit funny and Sippy has been lame
ever since.
