Not sure what you mean by "cut". SWR issues can cause all sorts of problems. Most modern communication systems will suffer from reduced output if a high SWR is encountered at the antenna. This causes the transmitter to "fold back" which protects the final amplifier circuit from damage in most cases. It also protects the finals from damage if the antenna cable should become open (infinity impedance) or shorted (nearly zero impedance). Commercial and ham radio rigs also use this type of protection. High SWR can result in getting RF into the stereo system, and causing transmit audio problems due to "RF feedback". Often these problems are brought on by poor or non-existant grounding systems. Make sure the CB is well grounded, and the antenna too, not just thru the cable. Grounded means TO THE FRAME at a clean non-painted spot. If your audio output varies when you turn your head, or you can hear humming or signal change in another reciever when transmitting, you are getting RF into your mic system. Again, this is USUALLY a ground problem. See if you can round up a ham operator or tech shop that has an antenna analyzer. (This is NOT the SWR meter that you find at the local truck stop) Ask them to check the SWR at both the rig AND the antenna. This should show if you have any cable problems. By careful and SMALL adjustments at the antenna, you should be able to get the SWR down to between 1:1 and 1:2.This should show if you have any cable problems. With a power/SWR meter in the system, Check and see if the SWR changes when you have your headset on and move your head around. This is another indication that you are getting RF into the mic system. Also, when the power meter is hooked up, see if the meter swings "upward" when you speak into the mike. Talk for a short while and see if the meter swing stays the same. This is a quick check to see if you have any failing componants in the transmit circuitry. CB radios are limited by FCC law to be 5 watts. Not much. In reality, they are more like a couple of watts or three or four at best. Any problems getting that signal, along with the modulation (your voice) to the radiator, cuts down on performance pretty quick.
If you mean "cut" by cutting in and out, then you have a loose connection somewhere.
BTW.... most coaxial cable sold with CB type equipment is pretty much crap, having very low shielding quality. Good RF coax is cheap, and can save you a lot of headaches. Many, many problems are the result of poor coax and poor grounds.