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"headway on" is a correct phrase that can be used in written English
It means progress or advancement in achieving a particular goal or making a change. Example: The team has been making steady headway on the project, and they are confident they will meet their deadline. In this example, "headway on" is used to show that the team is making progress towards completing the project.
Exact(60)
"But I think they're actually making headway on that one.
We are making some headway on cross-border co-operation.
Meanwhile, others are making headway on fund-raising.
The N.F.L. has made headway on college campuses, too.
In September, Marclay was back in London, struggling to make headway on the manga book.
So the revenue side isn't that bad; we do make some headway on unstarving the beast.
In the Democrat-controlled committee, conservative opponents made little headway on altering the bill.
He added that the opposition "is making headway on the ground".
"The other big question we hope to make headway on is what is the dark universe….
We're not making much headway on the grief industry, so I try internet porn.
That's why the Democrats are having a tough time making headway on the economic issues.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com