There are many, many variants of this surname in Ukrainian. I am clarifying only the ones of interest to posters in this thread.
Hawryluk - Hawrylyuk = Гаврилук-Гаврилюк. Currently 44,000 w/lyuk-liuk variant surname living in Ukraine.
Hawrylyak/liak = Гавриляк (Although rooted in same male name Hawrylo/Havrylo, the suffix iak/yak makes it a different surname.) Appx 3,300 Hawryliak living in Ukraine today.
Hawrycszak = Гаврищак. Appx 160 Hawrycszak living in Ukraine, mostly in Ternopil and general Podolia regions.
Hawrylyshyn/Hawrylyszyn = Гаврилишин. Mostly in Historic Galicia Province, w/greatest concentration in Ternopil and Lviv regions. The suffix SHYN/SZYN makes the surname distinctly Ukrainian. It is unusual, because it has a female root.
The key is to locate the EXACT PLACE OF ORIGIN.
Best to start w/Hamburg (exit port) Immigration Records. http://search.ancestry.com/
Ancestry Membership, even a temp one, will allow viewing the ship manifests, where the details can be invaluable.
Understanding what and where AUSTRIA was from late 18th century to 1918. It is not today's Austria. Short read makes it easy to understand. http://www.infoukes.com/genealogy/trac/baba.html