If Mr. X married Ms. Y and they got divorced and have a Get, and Mr. X continued to marry Ms. Z and is now asking her for a Get in order to re marry Ms. Y. Is that permitted?
In terms of Mr. X remarrying Ms. Y., that is permitted provided that Ms. Y did not marry somebody else in the interim. However, your wording of "is now asking her for a Get in order to remarry Ms. Y" is not so clear. First of all, he would be giving her a get, not asking her for one (perhaps you mean that he is asking her to accept a get). Secondly, I would recommend that Mr. X work on having his current marriage be a successful one, without planning what to do after it may dissolve.