The phrase "headway with" is correct and usable in written English.
It generally refers to making progress or achieving success with something. For example, you could say, “We’re making good headway with the project.”.
You would come home despairing at making headway with him".
Most of them are already making headway with pure electrics.
It has made even less headway with the Chinese.
The original director, Michael Anderson, made no headway with it.
Joyce had made even less headway with Ulysses, a work he had been planning since 1906.
I can claim no headway with Italy, the next G-8 host nation, either.
Among the things that companies have already made good headway with is liaising with teachers.
But these techniques have not made much headway with large manufacturers.
One Home Office source said: "He believed he made headway with backbenchers".
It was at this point that we really started making headway with her reading skills.
Handsome and mysterious, they made headway with leisurely thrusts of their fanned tails.
