Can he take his mother to the market when she is wearing a short abayah?

Firstly: 

You have to advise your mother in a kind and gentle manner not to wear this kind of abayah that shows the shape of her body, and to adhere to the conditions of shar’i hijab, one of which is that the clothes should be wide and loose. 

For more information on the conditions of correct shar’i hijab, please see questions no. 214 and 6991. 

Secondly: 

If your mother is going to go to the market in any case, whether you take her or not, and if you refuse to take her she will go alone, then in this case you should take her to the market to look after her and to reduce the evil action as much as you can. 

And Allaah is the One Whom we ask to reform the Muslims.