Hi,
The rental property in question could be treated as an FHL as long as it meets the full criteria here:
HS253 Furnished holiday lettings (2020)
We are unable to answer your main question in respect of the type of rental agreement that you have entered into with the management company. We suggest that you supply full details in writing of the actual type of agreement in place to enable HMRC to review.
Generally as long as the actual FHL criteria is met, then the income would be treated as being received through an FHL
Thank you.