Define value of a centroid I want to choose a clustering algorithm for which I can define the value of centroids (and not only the number of them).

Which algorithm should I look at?

My understanding of k-means is that I can specify 'K' (the number of centroids) but not the value itself.

Instead, I have a given number of centroids that I want to "test". Some of them will be indeed the actual centroid, whilst others will have only few to none values and therefore I will drop them.
However, I don't want to create new values for the centroids.
If you want to give the means, then it's not clustering anymore.

The name-giving part of k-means is computing the mean.

Instead, you are doing nearest neighbor classification. Your predefined means are your "training" data.