The screen control on the flyback adjusts the voltage applied to the screen grid (usually G2) in the electron gun of the CRT. It is typically on the order of about 400 volts and is derived via a divider from the HV in the flyback. The focus control is also generally there as well with typical focus voltages around perhaps 300 volts.
When you add a screen grid to a tube (which reduces the grid-anode capacitance, among other things) you need to bias the screen up from the cathode potential in order for the later anodes (modern CRTs often have multiple accelerator anodes, sometimes at different potentials) to be able to pull the electrons from the cathode, otherwise they would just fall back into the space charge around the cathode.
If you look at the gun assembly of a modern CRT (like after the 1940-50s and color CRTs) it will usually go something like:
Cathode(s) - G1 (control) - G2 (screen) - first accelerator - G3 / focus - 2nd accelerator - anode (the HV at the glass screen at the front with the phosphor)
Using the service information for your set, you generally follow a procedure like setting a specific bias level on your gun(s) cathodes or control grid using the gun balance controls or the brightness on a B&W or something, then bring up the screen until just before you see the raster appear, then adjust the biases using some procedure, but this varies between set manufacturer, design, era, etc.
What TV make and model is this that you are working on? That would make it much easier to give better specific information.
Edit: I see you posted a picture. On this model the screen and focus controls are on the neck board rather than the flyback transformer itself, obviously, but have the same function regardless of where the control is physically located.