Since your wrists are pretty weak introducing weighted exercises might be abit hazardous so here are some exercises us boxers use to strengthen our wrists so we can punch harder without hurting ourselves, if your not clear about anything let me know I probably won't be able to explain too well what I want to say:
Take one hand facing palm down and grab it with the other hand from the palm and fast but gently bend your wrist downards, try it out for a bit at slow speed and get used to it, make sure ONLY your wrist moves, nothing else must be at motion, you can do this as many times as you want but if you shoot for higher reps try not being so fast when doing it, pick a light weight up after you've done about 30 you should feel a small difference right away.
This next exercise is known as Kotegaishi, all your really doing is taking your right hand and putting it out, using your left hand to grab it and tightly and massaging it up and down(reminds me of something
) from the top of the knuckles to the bottom of the wrist, this one might surprise you so be careful, overdo it and it will hurt you
Next exercise is put your hands together like your clapping and push your arms in and out trying to shift as much movement into the wrists as possible, isn't very effective on its own but a nice addition.
This is a badass wrist exercise, what you do is stick your arms out, take your hand and turn it upside down and grab it with your other hand and then try and pull your wrist towards your chest, BE VERY CAREFUL not only is this exercise painful but its very tricky to perform correctly, if you can't do it without hurting yourself then don't bother till your much stronger, it might not be so difficult for you if you've got loose wrists which you probably have considering theyr small
Last exercise is Sankyo, its similar to the last exercise, put one hand with thumb down and grab it with your other hand and twist the wrist far as you can and back, again be careful.
These exercises are beginner and should get you on the way, I've got some high impact and weighted exercises too but I figured I'd never hear the end of it if you injured yourself so I went with the low impact ones