How to best represent missing count data?

I have BTO data within the region of Norfolk over the last 10 years.

For the counts, there are no values for the majority of the data, this may be because someone forgot to mention how many birds they saw in their garden, hence no recorded counts. However, they would still provide the identification of the bird, just no counts.

Is there a way to represent this statistically for inference? I have thought of filtering only the data with counts, however, I lose the majority of the data, which may be useful.

Help me understand the best statistical approach (using R), that may include the data with missing counts. Or is the best approach to filter the missing counts and stick with those with counts only?

The purpose is for understanding diversity in this region.