The Adapter-side logs regard custom code and we aren't able to find clues about the update flow.
Until we have a clear identification of the problem as the user perceives it, I don't think that we can proceed further.
If you confirm that the issue affects only a few clients and doesn't affect the other ones, it is more likely that the problem is related with the client application's ability to keep the pace of the incoming updates while trying to process them.
Please clarify if the client application's job is to visualize the data or to elaborate it internally.

BTW, the client log regards February 27, that is, the day before the Server and Adapter logs.
Could it be a mistake and perhaps the issue was observed on February 28 and never on February 27 ?

The refusals don't seem to affect the performance. Only in case a client application gets several refusals before succeeding, this may be seen by the user as a delay. Until we have a clear identification of the problem as the user perceives it, this is also a possibility.
Or perhaps this is just the evidence of another issue to be investigated.