LEO already has aufs Geratewohl and aufs Glück for "on the off chance". It doesn't seem necessary to include "sich aufs Geratewohl bewerben," as there's nothing special about the combination as an idiom: the meaning can easily be deduced from the existing definitions of to apply = sich bewerben and on the off chance = aufs Glück/Geratewohl.
If the entry is maintained in LEO, however, then the English side should be changed to something more general, like "to apply on the off chance of success." There is nothing in the German as to what kind of an application one is making aufs Geratewohl; one could be applying for a scholarship, internship or membership in an organization.
Also, the sources I checked, both AE and BE, are universal in giving the unhyphenated form as correct. The links are included above. If the entry is kept, it should also be corrected from off-chance
to off chance
, which would match the other entries in LEO, which are not hyphenated. Dictionary: off chance