How does potentiometer influence incoming current in the circuit?
The current in the circuit will be determined by the total resistance in the circuit.  If the potentiometer is set to 1000 ohms, the total resistance will be 1100 ohms, so, with a 4 volt power supply, the current will be 3.6 mA (I = E/R = 4/1100).  If the pot is set to 2000 ohms, the total resistance will be 2100 ohms, so the current will be 1.9 mA.  (Your numbers may vary a bit, depending on how accurately you set the pot, and on the resistance of the ammeter).

Kirchoff's Current Law says that in a simple series circuit such as this, the current is the same at all points in the circuit.