She had great difficulty finding a job.
UK: People with asthma have difficulty in breathing.
It seems that British English puts an extra "in" there while
difficulty + doing are acceptable both in AmE and BrE.
Is my understanding correct?
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No, your understanding isn't correct. It's not really that there's a usage split between British English...
...and American English...
It's just that usage has changed over time (on both sides of the pond). I see no significance in the fact that the "crossover point" (when it became more common not to include the preposition) was apparently a decade or two earlier in print in the US than the UK.
I can't think of any contexts where it would actually be "incorrect" to either include or not include in between difficulty and the associated gerund / continuous verb, and I doubt many native speakers would notice if you always made the "less common" choice with that particular noun.
But with closely-related forms, such as He had a problem [in] starting his car, the unwanted preposition would be very noticeable, so I suggest learners avoid it in all such contexts.