Sleep Data from Apple Watch and Pillow in the Health App

Hi

I've just bought an Apple Watch and am really disappointed with the Sleep function - I find it incredibly inaccurate. I've since downloaded Pillow, which seems to be better and recognise the difference between being in bed and asleep. However, when I open the Health App, the Sleep section is still showing the inaccurate data rather than the data from Pillow.

All the right permissions are on and the Health App is syncing with Pillow to get the data but it's like the inaccurate Apple Watch Sleep data is overwriting it. How do I get the Pillow data to show?

Thanks