'뭐 하러' means 'in order to do what' and '뭐' is a colloquial form of '무엇'. cf. '뭐하다' is short for '무엇하다'. '무엇하다' is an adjective which means 'to be awkward' or 'hard to say'. In 2017, National Korean Language Institute revised the Korean dictionary and they removed '뭐하다' from the list of 'verbs'. So there is only one way of writing or using '뭐 하러' with a space in between '뭐' and '하러' is correct. Again, only '뭐 하러'(with a space) is correct.

in Korean spaces are used to sepatate words