"fills the gap" is correct and usable in written English.
You can use it to refer to something that fills the space between two other things. For example, "The new policy fills the gap left by the old one."
Sumner gamely fills the gap.
National Service fills the gap commendably well.
Whenever we have downtime, the Internet is an enticing, quick solution that immediately fills the gap.
Mr Mitchell fills the gap with a number of clever, if somewhat disconnected set pieces.
Financial engineering fills the gap between people's desires and their wherewithal.
It is often the case that the community fills the gap left by social services.
How long will I have to wait before the replacement fills the gap?
Essentially, it fills the gap between a purchase price and the value of the assets acquired.
In New York and California, generous financing for AIDS programs fills the gap for hospitals.
The subject has been rather neglected by academics, but Vinen fills the gap brilliantly.
