log shipping backup fails every so often on only one server
Since your network drive is available most of the times and the backup job only fails randomly, I would first check the network. It could very well be the network issue. I would also look at the intervals the backup job fails, to see if there is a pattern to it.

Coming to restarting the logshipping part - the easier method is to restore the full backup and the standby server and reconfigure. But it depends on the size of your database, resources available etc