I don't know of any PerlDesk clones (especially as I've never used PerlDesk), however there are a number of excellent Help Desk Support/Ticket Tracking tools and applications available.
My personal preference is Best Practical's RT (Request Tracker). Its primary interface for users is via e-mail, so they don't have to worry about the tool involved. To open a ticket, they send an e-mail to a support address, and a ticket is automatically created. A web interface is also available for users, and is used by support personnel.
RT is Open Source, and actively developed.
Others you might want to check out include OTRS, and possibly some of the more developer oriented bug tracking systems like Bugzilla. They can be used for issue tracking, although not all excel at it.