From what I've read if an access switch like our HPE have only an uplink, configuring STP on them is not really needed. Am I wrong?
Depends on what's possible regarding accidentally creating a L2 loop and your risk tolerance.
Something like STP is really needed for by design L2 loops, such as the core ring, for redundancy, design.
As usually, switches, even L3 switches, have multiple L2 ports in the same L2 domain, something like STP avoids the impact of an accidental loop creation.
Ditto for multiple switches sharing the same L2 domain.
Your HP and Catalyst switches, I understand, share L2 domains, but if there's no way to create a second inter switch link, they don't need to share STP, but STP still highly valuable for handling accidental L2 loop formation on just the HP switch.
Does the forgoing help?
