You are correct, it depends. Coax cannot be spliced but you might be OK to repair the outer insulation /if/ the shielding is not compromised. The very outside layer of insulation just protects the conductors and keeps them in place. If you just nicked it off, wrap with tape.
If any of the shield conductors are cut or displaced, you have a choice. Replace the length or cut and splice in connectors. Coax has a uniform impedance of 50 ohms and any splice would be a major disruption, a mismatch if you will. SO239/PL259 connectors do not have a constant impedance but are better than trying to cut and splice.
The right way is to replace the coax.
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