there are a couple of approaches, both of which involve stealing functionality from the old app and replicating it either outside or inside your new app.:
The first would be to build an http4k proxy on the outside which just passes through traffic to the spring app. then gradually steal the traffic from the spring one until you are done.
The second would be to build the http4k stack inside the spring app and reconstruct http4k requests/responses in the controller to be passed through, injecting all of the dependencies for each endpoint (which can be magically injected into the spring controller). then you end up removing the controller itself.