When I wrote  "I don't think you have one, but then you need to make one", I was not referring to the number itself, but a link per photo to a place where you explain the number. 
The reason you need this is because we have said that the unique identifier is necessary to participate in WLM. During the WLM contest lots of mistakes get uploaded and many end up getting deleted because they cannot be traced to the proper record.
I don't know if you can link to your wikipedia list. Usually this link goes directly to a heritage agency.

Jane, our Ministry of culture (according to our law, a list of cultural heritage is its responsibility)  do not publish a list of monuments in its page. Of course a link to the Ministry of culture page is possible but it is not useful - one can read nothing about cultural heritage here. The only possible link to a page with an explaination of the unique number in our case is a link to WLM-project pages with a list of monuments with unique ID from each one, displayed in a WLM-row.

In this case I think it would be more reasonable to link to WLM-project and explain that the number is given for WLM-competition. Maybe instead of Ukrainian flag it would be better to place WMUA logo or something like that?

Andrij, you might want to edit those lists to only include numbers for the ones eligible for photographs

As far as we have lists of monuments in a main namespace, I suppose there is a reason to include to competition lists eligible monuments only. What do you think about it? For example lists of Kiev monuments would be pretty cut.

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Andrij