Addition of a Person as an Accused.

Yes a person can be added. Please refer s.319 of Cr.P.C.

Section 319 comes only after cognizance of offense and not before that.

Good morning Gopal. the answer is No because there is no chance to the Magistrate to apply his mind before issue of process. The process of cognizance was now reached at the stage of making the mind to issue the process. The right is given only U/s 319 is right from AK ji and RKji. What it says? S.319 comes into operation only in the course of inquiry or trial.Bhola Rai v State of Bihar, (1997)3 Crimes 48 (Pat). 

Under section 319 Of code of criminal procedure , court can  add aaditional accused at any stage , but this is only dicretionary power of court.

Please note that query was "after taking cognizance of an offence".  In any case process is issued after taking cognizance. Taking cognizance and issuing process is the stage of inquiry.  Hence s.319 Cr.p.c. squarely applies to the query raised.

s.319 of Cr.P.C

 What happens when process has already been issued, trial has started and then a person is added as an accused? 

yes.mr.adinath answer is very clear.