Can I wire a Pump Start Relay to an outdoor GFCI outlet?
This is why GFCIs have LOAD terminals

A receptacle-type GFCI has two sets of terminals: LINE and LOAD.  The use of the LINE terminals is simple -- these are where power is applied to the GFCI.  The LOAD terminals, though, are interesting -- they are effectively a third receptacle on the back of the GFCI, only implemented using screw terminals or backstabs instead of normal NEMA receptacle contacts.  In other words, they extend GFCI protection to whatever is connected to them.

So, you can connect the wires feeding power to your pump start relay box (hot + neutral) to the LOAD terminals on the existing GFCI.  This way, you don't need to put in another GFCI, just a standard receptacle, and you don't have to worry about oddities that might be caused by the second GFCI having power cut to it on a regular basis.  Just remember to put one of those "GFCI PROTECTED" stickers on the regular outlet that you're wiring up to the pump start box if you do this!