First let me start with Khwahish, or Khwaish as it is spoken sometimes, which is not entirely correct. Khwahish is simply a wish or desire. You ask for something or not, if you wish to have it, it's a Khwahish.
Coming to Guzaarish, it's a request, a petition. You can say that doing a guzaarish is to plead in the earnest.
A similar looking word, Farmaish comes there too. The word means request again, but Farmaish is used in a slightly different sense generally. Farmaish is asking for something from somebody who can or generally does provide it. Also, you may not be asking it very needfully. If you ask for a biscuit from your mother, it's more of a Farmaish than a guzaarish. Requests to artists, like a request to a singer to sing some particular song, are generally called Farmaish too. A farmaish, in general is not as serious as a guzaarish.