Manually control smartphone power current drawing during charge?
It depends on how a phone implements this power adapter detection. Some charging circuits have their own microcontroller and software which you have no access to.

Depending on the phone it might be possible to "let it think" there is a more powerful power adapter connected simply by shorting the two USB data lines (usually a white and a green wire) in the charging cable.

There are also other detection mechanisms which work with certain voltages being present on those same data connections.

This all assumes that the 5 V USB voltage is at least somewhat stable. If you decide to use a charger which cannot meet this requirement then all bets are off, you should really use a better power adapter or accept the slow charging.