I tried to put the same kind of desk fan at the window, one way and then the other, for a few hours, to see if it had any effect. It was a very hot day but colder outside than inside. The building's concrete was likely still radiating the heat from the day before and there was no wind.
I see now that my observation at the time was right: it did nothing to the temperature, and it might have worked better if I had put the fan 1-2 meters away from the window, directing it towards the window. Now, whether the effect would have been significant anyway… we'll have to wait for next summer to know, I guess. I'm not particularly looking forward to it, though.
An easy mental model is imaging the air is water. Close a door on a room and it'll fill up and block the hose.
PS: a box fan and a 5" thick MERV13 filter makes a heck of an air filter. 2" likely also will work. MERV13 is great, but some HVAC can't handle it, and it takes a couple passes (term is air exchanges per hour, I think) to capture what HEPA does in a single pass.
The fan increases air speed at the centre of the rotor, creating a low pressure zone which then sucks in surrounding air. So it helps to place the fan away from the window (roughly far enough that the wind cone "fits" the opening).