Worn Chainring repair

[Piedwagtail]

Hi everyone,
Noticed the other day that my smaller ring is getting very worn, i hardly use the big ring so it still looks fine, but i then forget to look behind it.
The chain is sitting on top of the teeth rather than between them, so i was thinking as a temporary measure, could i file each tooth down??
My logic tells me that it might lets the chain sit a little bit lower, then i might survive the winter without forking out money on a new chain set so i can save for a new bike next year.
Anyone done this sort of thing before?? Or just recommendations on whether it could work??

Thanks in advance
Jobe