Some HSMs allow you to run custom code (usually a co-processor within the HSM’s security boundary but distinct from where the keys are loaded / raw crypto primitives happen).

Other HSMs don’t have this ability.

Finally some vendors will tell you that their hardware is extensible by them implementing custom logic/algorithms.

If you want to play around with HSMs (or any enterprise-grade hardware) I would recommend looking for used stuff on eBay or talking nicely to sales rep who might lend you a demo/eval unit.