As you have mentioned, one of the little plastic nose-protector things on your glasses is breaking. You can go to the glasses stores where you buy your glasses and ask the technician there if the broken plastic nose-protector things can be replaced. If the answer is no, then you have to change a new frame.
They are easily replaceable. You don't need to buy frames if there isn't other damage on your glasses. You can ask an optician to replace them for you or you can replace them by yourself if you have a pair of new nose pad and some small tools.
In most cases, the nose-protector can be replaced because some sellers will offer you some spare ones. If you do not have spared ones, you can take your eyeglasses to your local optical store to get it replaced.