What is the small "filter" in my ear hooks?

The technical answer is that they are used to smooth the frequency response of the hearing aids.

This means that the BTE style hearing aids naturally have some peaks in the sound level at certain pitches. The filter helps to make the sound more even across the different pitch (frequency) range.

There are fewer and fewer of these filters in today’s hearing aids. The smoothing process can now be managed on the sound processing chip.

I would say you are looking at a wax filter.

Thanks, I lost one and the sound of the aid is definitely different. Louder and more “tinny”.
Sorry for my lack of better science but “tinny” like in tin is all I could come up with.

They are - as already mentioned - a sound damper.

They are porous and are made of sintered metal or ceramic powder.

This material can become damp or oily and so can block up, thus blocking sound.

Changing the hooks can fix the problem - or if it happens a lot use a hook without the damper … but tell the fitting software too!