An ASP is regarded one of the most powerful police officer in hierarchy. He at most places exercise the powers of dpo.
However, ASP can award minor punishments to constable and head constable like deduction of salary, quarter guard etc.
Kpk has amended the police rules and now asp/dsp can give minor punishment to asi as well. DPO has the authority to give minor as well as major punishment upto the rank of SHO. Major punishment includes reversion, dismissal from service, reduction in rank etc.
But keep in mind if ASP recommends to dpo to punish some official, in 99% cases dpo acts upon his recommendation.
These are powers with respect to discipline.
With regards to general policing work, asp has all the powers which an sho has in a police station (as mentioned in sec 551 crpc).

__________________
Whatever mind can conceive,it can achieve - W.Clement Stone