Can one still read Tahajjud prayer if the Fajr time has started ?
Example starting time for Fajr on calender is 6:30 am and jamat time is 7:30am.
Yes, but before the Sunnah of Fajr. This is a dispensation (rukhsa) to preserve a good habit as the strict view is that after adhan the only voluntary prayer is the 2 raka`ats of Sunnah.
Hajj Gibril Haddad