The problem you're having sounds like it's related to the last item in the V4 miniscope assembly guide:
The first thing you need to do is solder the coaxial cable (the white line) to the U.FL connector, which is described in more detail on the github page. They also have some nice diagrams.
If you have the coaxial cable (the white line) soldered to the U.FL connector, then you have to connect the U.FL connector on the cable to the corresponding connector on the miniscope. The two connectors should look something like this:
The connector on the cable should look like the piece on the left, circled in black, and the connector on the scope should look like the piece on the right, circled in white.
In this picture, the two connectors are not attached. To connect the two pieces, you just have to pinch them together. The metal ring on the scope body fits in to the black plastic socket on the cable connector, though it make take a fairly firm pinch to get them to snap together. It's possible to bend the fins on the outer edge of the cable connecter, which can make it hard or impossible to fit the two connectors together, but you should be able to bend a warped fin back into place without too much difficulty, allowing you to try again. Once they're connected, it should look like the original picture you attached.