There is probably corrosion and/or grit inside the pawl mechanism. Inexpensive chinese freewheels are notorious for this. You might be able to pull it apart, and if you have the patience of a saint, even get it back together again! Is it worth the time and trouble on a $20 freewheel that wasn't very good to begin with?
Considering how cheap a new freewheel is, as well as the vast number of slightly used decent freewheels available, I wouldn't bother fixing the old one. It's toast-toss it. Replace it with a decent one, not the first chinese piece-of-$h!t they try to sell you. Those chinese freewheels are made solely for the department-store bike crowd. If the local shop doesn't carry SRAM, go to one that does, or look on the used market.