Yes, you can splice it.
There are some RG-58 splice connectors out there, but I haven't seen them in a while.
- found one, here they are:
Amazon.com: SPLICE CONNECTOR for RG8X & RG58AU: GPS & Navigation notice the low reviews...
For higher end cable, like 1/2" heliax hardline, you can purchase splices:
https://www.tessco.com/products/displayProductInfo.do?sku=95572&eventPage=1
Here's why you probably shouldn't do it, though:
The cheap splices for RG-58 aren't very reliable.
RG-58 is cheap cable, it's much easier to just replace the cable.
If not done right, you can get an "impedance bump" which can cause issues.
For a CB antenna, were talking a couple of watts. Unlikely, if done right, you will damage anything. A small impedance bump isn't going to be the end of the world for to a CB.
You can get a crimp or solder on connector for the RG-58, in fact you can get a male connector for one side and a female connector on the other and just connect them together. A properly installed connector (or two) isn't going to be a big deal. While all connectors have some sort of loss, it's usually on the order of 0.1dB or so on a properly installed connector. I'd challenge anyone to hear a 0.1dB drop in signal without using test equipment.
But seriously, take the antenna apart and install a new cable. Route the new one so it doesn't get damaged, or better yet, do a permanent install once and for all.