What's this piece???

Looks like the little filler sponge that goes between the short sides of the pint head and the plastic ramps on each side.
Been a while but when I had a 1304 as part of the cleaning kit it would come with 2 new ramps and the foam was stuck on each.
Either that or check your spitting sponge(if your printer has them) to see if it is coming apart.

 

Not sure how critical they are. I think they soak up the excess ink when a cleaning cycle is run.
Maybe just swab around the areas after you run the cleaning so nothing drips down on the prints.
What model of printer do you have?

 

That is the absorber material between the printhead and the plastic ramp

 

I don't even have those on my brand new 1204 head. Don't worry about it - get rid of them. One more thing to grab and hold ink and misc fuzz...