Can a 555 be used to control the charging and discharging of a capacitor to provide bursts of current?
why would you use a NE555 for something like that? You just want to switch on a transistor above a certain voltage threshold, and then later switch it off as soon as your voltage falls below another level.

In other words: you want a Schmitt trigger:



you could actually simply copy the circuit from the above linked wikipedia article:


V_ would be your ground, V+ would optimally be some kind of stable voltage source (e.g. a linear/low-dropout 3.3V regulator) and Vin would be the voltage across a voltage divider. Vout would then be used to switch a MOSFET:



You can follow the method described in the wikipedia article to select RC1, RC2 and RE, and still have the freedom to select Ri1 and Ri2 to "fine-tune" your input range (e.g. use a trimmer/poti to "prescale" your voltages).