How to keep a component "pressed" so the chain remains hooked? I am really at a loss of words to describe my problem, so apologies in advance for the poor choice of words in the title.

Basically I have this component at the rear wheel (which I think is called rear derailleur):



and how far it is from the spokes of the wheel is controlled by the springs(?).
The trouble I have (as it seems) is that this component is too "loose" causing the chain to not be able to catch on to the component's "wheel" properly.

Compare:

Not pressed:



Pressed in:



And notice how in the former case, the chain doesn't lie flush with the "component"'s metal wheel. So that when I cycle the pedals, the chain always unhooks and causes a jam.

Is there any screw etc. that I can adjust to keep the component "pressed in", presumably something to do with the spring..
Yes, that is the rear derailleur. FYI there is a Terminology Index on this site that provides all component names.

It seems the problem you are having is that the derailleur is not lining up with each of the rear sprockets, so the chain is not running on them properly.

Most modern derailleur systems are indexed, that is the shifter lever has discrete clicks that correspond to each sprocket. What has probably happened on your bike is that the cable has stretched or loosened so that the derailleur does not sit directly over a sprocket at each position.

The derailleur or shifter cable usually has a barrel adjuster that can shorten of lengthen the cable to adjust the dearilleur position. In your case the cable likely needs to be detached from the derailleur and the effective length reset.

There are several online guides that detail how to adjust derailleurs. This one from Park Tool is probably one of the best ones - https://www.parktool.com/blog/repair-help/rear-derailleur-adjustment.