This is normally a divide and conquer problem, although on a cheap bike, it could be the sum of a number of small problems - if the latter, be prepared to have your patience tested.

First instinct is to suspect the cable, but it could be elsewhere. That said, first place I would look is the derailleur hanger and see if it looks straight. If obviously bent, deal with that. If not obviously bent, then move on to cable (but not completely excluding hanger).

Next look at the cable. Look for obvious pinched outer and tight radius turns - anything that would stop a new cable sliding easily. If this was an old bike, I would suggest lubing the cable at this point. Given its a cheap bike, lube is perhaps worth a try.

If still not obvious problems I would cheap the hanger alignment with a tool (I left this last as I presume you do not have the tool, and it requires removing the derailleur.

Last step is probably to replace the cable. I would suggest both inner and outer as if the cable is at fault, it is likely the outer has something like crunched up liner.

If still no obvious cause, remove the wheel. Check derailleur spring tension first (Hard to explain how much it should be - enough to firmly pull the cable). Check the feel of the cable sliding though the outer as you move the derailleur.

Shifter is unlikely to be cause of the problem. They tend to work or not, but really cheap once can have problems. While I could check a shifter with reasonable confidence myself, I have no real idea how to describe what would look for.

This is normally a divide and conquer problem, although on a cheap bike, it could be the sum of a number of small problems - if the latter, be prepared to have your patience tested.

First instinct is to suspect the cable, but it could be elsewhere. That said, first place I would look is the derailleur hanger and see if it looks straight. If obviously bent, deal with that. If not obviously bent, then move on to cable (but not completely excluding hanger).

Next look at the cable. Look for obvious pinched outer and tight radius turns - anything that would stop a new cable sliding easily. If this was an old bike, I would suggest lubing the cable at this point. Given its a cheap bike, lube is perhaps worth a try.

If still not obvious problems I would check the hanger alignment with a tool (I left this last as I presume you do not have the tool, and it requires removing the derailleur.

Last step is probably to replace the cable. I would suggest both inner and outer as if the cable is at fault, it is likely the outer has something like crunched up liner.

If still no obvious cause, remove the wheel. Check derailleur spring tension first (Hard to explain how much it should be - enough to firmly pull the cable). Check the feel of the cable sliding though the outer as you move the derailleur.

Shifter is unlikely to be cause of the problem. They tend to work or not, but really cheap ones can have problems. While I could check a shifter with reasonable confidence myself, I have no real idea how to describe what would look for.