I would like to know if it's possible to do a PhD when I'm in the UK under the HSMP program

hi Alisa, if you'd like to do it part-time whilst working full time, then it shouldn't be a problem. But if you want to do it full-time, then you'll have to reapply for the FLR(S)to switch to student visa. The HSMP visa is equivalent to a two-year work permit which allows you to work, but not to full-time study. You must apply for switch before your programme starts. Hope this helps.