opencall.Priority = 
averages.Where(x => x.ProblemCode == opencall.ProblemCode)
.SingleOrDefault().Priority;
The above lambda statement returns some nulls because ProblemCode isn't always guaranteed to be in the averages list.
How can I rewrite this statement so that if that is the case opencall.Priority is set to "" instead of the application throwing an error?
You have to provide a new default value for your reference type, other than null.
opencall.Priority = averages.Where(x => x.ProblemCode == opencall.ProblemCode)
                            .Select(x => x.Priority)
                            .DefaultIfEmpty("")
                            .Single(); 
So Priority is a string? Note that you don't need SingleOrDefault anymore since the query can never throw an exception because it is empty when you provide a DefaultIfEmpty.
Split it up:
 var result = averages.Where(x => x.ProblemCode == opencall.ProblemCode).SingleOrDefault()
 opencall.Priority = result != null ? result.Priority : string.Empty;
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