How can I secure a loose kitchen cabinet?

Generally speaking, cabinets like this are mounted on the back panel. That is where the weight is carried. Trying to pull this cabinet sideways is an improper fix. I'd be willing to bet that it has pulled away from the back wall slightly, which is the primary problem. The thin filler strip along the side is fit and installed later, and doesn't really bear weight there. It can add some support, but it's secondary to the strength of the back panel.

You need to investigate all the ways that this cabinet is mounted to the wall, then find out whether any of the back panel screws were left loose initially, or have come loose, or if the structure of the cabinet itself has been compromised.

If you do find that situation, the first step is to get the cabinet back in position. You can do this with temporary legs under the front corners, or by having a helper push the cabinet back tight against the wall. Then you'll want to add new screws, or at least snug down existing screws. If the cabinet is coming apart, it almost certainly has to be taken down and repaired.

Be sure all screws are installed in locations intended to bear weight. Some cabinets have a very thin back panel with thicker rails at specific locations like the top and bottom. The screws should have wood threads and should penetrate framing members 1 to 1-1/4". You might use one of the existing screws as a length gauge, though screws that were too short may be part of your actual problem here. You'll have to know the details of your wall's construction to be sure.

Note that modern cabinets are built almost exclusively with screws, and they are mounted to walls with screws. There's no hammering involved with cabinet installation. As a general rule in carpentry, if nails have come loose, pounding on them won't solve the problem. They'll stay loose and you're likely to wreck things.

What you have there is an interior construction screw. They have a smooth shaft near the head so the initial piece of wood they go through doesn't have any tension force or shear force applied by the threads. This lets you screw it down tighter compared to screws that are entirely threaded.

The most important part: I know it is tempting to just take a driver and tighten it, but doing so will likely destroy either the cabinet or the drywall.

You should first take everything out of that cabinet, the one below it, and the two above the fridge. Then, get on a good stable step stool or ladder and try pushing the cabinet towards that gap from the inside. It should, with a little force, slide smoothly against the wall. If it does, try screwing it back in while either you or a second person hold the cabinet tight. Don't overtighten, it just needs to hold the cabinet even, otherwise you'll drive it through the wood.

If it doesn't slide smoothly, we'll need a few more pictures of the surrounding cabinets and how they're mounted to better judge the situation.

Frame challenge - I think all you need to do is adjust the hinges.

As you have re-drilled the hinge positions, they will be slightly different to how they were before.

Most hinges supplied for cabinets like this are adjustable so that you can slightly change how they sit against the cupboard and in relation to the adjacent door. Exactly how they are adjusted depends on the brand, but it is usually 1-2 screws that allow you to adjust in the X-Y direction so that the door aligns with the adjacent, and a 1-2 screws for the Z direction so that it sits close to the cabinet itself.

For instance, here's a popular brand in my country: Häfele (no endorsement, just readily available worldwide I think). In the image below you can see two screws on the long narrow bit to the right. The right hand screw on this (rear of hinge) gives you adjustment of how close the door sits to the cabinet, while the left gives you door angle. The corresponding base-plate allows vertical adjustment to align the doors.

You do need to adjust both hinges on the door at once.

Look up a manual for the brand of hinges you have, and it should come with instructions on how to do this, though they may be slightly opaque in what they mean.