[JRTJH]

You've got a broken/malfunctioning anti-siphon valve. It's the component in your photo at the top arrow.

The black tank flush system is not "connected to the plumbing system. It's a separate line that goes from the connection on the side of your trailer, to that anti-siphon valve, then down to the spray nozzle in the tank. It doesn't interface with any of the fresh water plumbing (other than sitting beside it under the vanity).

You can replace the valve with a similar one (available at any RV parts store, EBay or Amazon) or upgrade to a more reliable anti-siphon valve. They are available at Lowe's or Home Depot in the irrigation section. You'll need to also buy fittings to attach that valve to your current fittings where the old valve was removed.
Here's a photo of a typical black tank flush system.