czu001:
The box next to it on the left is a pololu voltage regulator that will boost the 3.3V to 9V. (Pololu - 2.5-9V Fine-Adjust Step-Up/Step-Down Voltage Regulator S9V11MA)The box underneath is a pololu brushed dc motor controller. (Pololu - BD65496MUV Single Brushed DC Motor Driver Carrier)
I am trying to boost voltage from 3.3V to 9V and pulse the solenoid open via the capacitor+resistor.
To pulse your 4Ω solenoid for 10 ms, while the voltage drops from 9V to 6V, you need a 2,200 µF capacitor. For a 50 ms pulse you need over 11 mF of capacitance. This means that your boost converter will have to supply the full 2.25A (9V/4Ω) for at least a few ms as the voltage drops, and that means it will try to draw nearly 7A on the input side. I was using this calculator for the capacitor.
As it's an inductive load the current is not instantaneous, that will help lessen the load and you to get away with a somewhat smaller cap than calculated, still it won't be long before it does reach full current.
You say you try to charge the capacitor via a resistor (I suppose to limit the current), but I don't see that in your circuit, only a 470Ω one in parallel to the capacitor (doing nothing more than wasting power that way).