I recognize those alligator clips — they’re the MA 1/MA 2 series from SKS Hirschmann and accept 2mm banana plugs. (There’s also a version that accepts 4mm banana plugs, as well as a crimp version.) The MA1/2 accept both old-fashioned 2mm pin plugs (where the pin is smooth, and the spring is in the jack) and 2mm banana plugs, where the spring is in the plug and the jack is smooth.
It looks to me like the lantern spring on the plug got a bit out of whack. As ajb said, the gap is supposed to be there — that’s how it gets mounted onto the shaft. I agree that it may have gotten a bit mangled. In the third picture, you can see the spring extending away from the shaft -- that's what's snagging and preventing insertion. On lantern-style banana plugs, the lantern spring normally sits inside a recessed section of the shaft, so that the ends of the spring itself are flush. On yours, as seen in the image where you circled the gap, the sprint appears to just be applied to a smooth shaft. Very odd.
It’s possible you could use smooth-jawed pliers to carefully squeeze the lantern back into round, but my hunch is that it wasn’t great quality to begin with, and that that’s why it failed.
Or you could just cut off the plug and attach a new 2mm banana plug. Just get one from a top brand (Hirschmann, Stäubli, etc.).
Find attached (in an reply I'll make in a moment from my phone) pictures of two styles of 2mm banana plugs from Stäubli, both as whole connectors and just the contacts within (shown in normal light and with raking light to highlight the gap). You can see the recess. Another picture shows regular-size 4mm banana plugs from Stäubli, in normal condition and with the lantern spring removed to show the recess it sits in.