He (broke)/(had broken) his leg in March and had been in hospital for about five weeks.
I think it should be (had broken) because it happened before he went to hospital. Is it correct?
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Either is possible. The choice does not depend just on the temporal sequence, but also on how the speaker is choosing to characterise the temporal relationships.
The and had been in hospital for about five weeks is located in the past relative to the narrative time, because of the past perfect. The verb in the first coordinated sentence could be located in the same way, but does not need to be.
As far as I can work out, this is general: I can't find a property of the particular verbs or circumstances that licenses this. I think it is because of the narrative convention that verb and verb normally represents a sequence.
"He BROKE his leg in March and WAS in hospital for about five weeks" is correct.
"He BROKE his leg in March and WAS in hospital for about five weeks before being released" is correct.
"He BROKE his leg in March and HAD BEEN in hospital for about five weeks WHEN HE WAS released" is also correct.
Past Perfect: When something happened, something else had already happened. The past perfect is NOT used to show that one past event preceded another, (e.g, I WAS born in Nashville and my father WAS born in Amarillo.)