OK, that is a universal aftermarket speed control that is part of the aftermarket control cable. You may not be able to get a perfect result with it.

The non-preferred way to adjust the cable, is to move the cable-outer where it clamps to the governor plate:

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Set the speed control in the Stop position. Holding the cable outer near the clamp (in the red oval in the picture), loosen the clamp screw and pull the cable outer about 2 mm to the right, then retighten the clamp screw. Push the speed control to the Choke position, and check whether the gap between the adjustable stop screw and the speed control lever is now zero (see green circle in this picture):

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If the gap is now zero, check whether the choke is fully closed (black bellcrank fully anticlockwise). If both these conditions are met, check whether the engine stops when the speed control lever is moved to the stop position (I'll explain how to do that in a moment). If the first two conditions are met and the engine stops in the stop position, the job is done and it is time to see if the engine starts. If the first two conditions are met but the engine does not stop in the stop position, move the cable outer in the clamp back 1 mm so it is only moved 1 mm from its original position instead of 2 mm, and repeat the tests.

Here is how to check whether the engine will stop when the speed control lever is put into the stop position:

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The object in the green oval is the kill switch: when its contacts are closed, the engine's ignition is disabled and it stops. It is operated by the speed control lever through the spring, connected to the rusty brown vertical lever on the switch. With the handlebar lever in the Stop position, the aim is to get that vertical lever on the switch to only move to the left until it is vertical - currently it moves somewhat further to the left than that. The best way to check the operation of the stop switch is with a resistance meter, but I'm just guessing you don't have one. If you do, say so and we'll use in instead of the "get the lever vertical" approach.

Remember, if you are faced with an engine that is running and you don't know how to stop it, there are two options. Option 1, just turn off the fuel and go have a cup of tea. The carburetor's float bowl will empty in two or three minutes and the engine will stop with no harm done. Option 2, pull gently on the spark plug lead's end cap (attached to the spark plug) until it pops off the spark plug, and the engine will stop immediately. However if you are going to use that approach, practice it ahead of time with the engine not going, or you may damage the ignition system (by pulling the lead out from the electronic module, instead of just unclipping it from the top of the spark plug).

If you have questions about this, let's discuss them before you make an attempt that ends up with the mower damaged or somebody injured.

The person who modified your mower seems to have a lot to answer for. The effects of using an el cheapo universal speed control cable instead of an aftermarket one which at least followed the Honda design, were aggravated by drilling a hole in the speed control lever on the governor plate to accommodate the quite different end fitting of the el cheapo cable. He drilled that hole in a poorly-chosen place in the speed control lever, which is part of the reason the cable inner is too long. In your picture it looks as if the cable outer is not even clamped at the governor plate end, because the over-length cable inner made that impossible. The easiest solution now would be either to drill another hole in the speed control lever in a more sensible location, or to fit a spacer at the cable clamp, to hold the cable inner further to the right of the clamp. We may have to discuss those options after you try to adjust the cable as explained earlier in this post.