What's the difference between the two? Example:
The classrooms had only students. Had the school a shortage of teachers?
The classrooms had only students. Was the school having a shortage of teachers?
(Or maybe one of them is ungrammatical?)
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The progressive expresses a continuing state rather than an event, so we ordinarily do not use verbs whose meaning designates a state in the progressive. It's superfluous.
Have is freely used in the progressive when it expresses an event (I'm having some people over tonight. or I'll have my secretary send it to you.), but in your example it expresses the state of undergoing a shortage. We would use the progressive, Was the school having a shortage?, only to suggest that we had a temporary shortage in mind.
Note, by the way, that the practice of inverting have with its subject in questions when it is not used as the perfect auxiliary has become quite rare, even in British English; it has virtually vanished from American English. Most people today would say
Did the school have a shortage of teachers?