A damaged covering does not stop the line working, but it removes protection at that point. Where the conductor is exposed or the covering is cut deep, a branch or a bird touching that spot can cause a fault just as it would on a bare line. Surface damage can also start tracking, and a cut lets water reach the conductor.
APAR's covered conductors use a longitudinally water-tight conductor, which limits how far moisture can travel along the strands from a damaged point. Damage should still be repaired, using a repair sleeve or insulating material approved by the accessory supplier, applied with the line dead and earthed under the utility's safety rules.
Most damage happens during stringing or when bare-line clamps are used. The MVCC accessories and installation guide covers how to avoid it, and conductor details are on the covered conductor page.