Is it possible to claim an incarcerated person on tax return?

If you are referring to claiming an incarcerated person as a dependent on your tax return, the answer is No.  The state provided over one-half their support, not you.  So that prevents you from claiming them as a dependent.

No.    A person who is incarcerated is being supported by the corrections facility that they are being housed in.  The money you may be providing for incidental expenses is not "support" that is sufficient to claim them as a dependent.  That money is gift money, which is not deductible.