Approach 1: Threshold the image such that only the black pixels remain on a white background. Smooth the image with a gaussian kernel/filter, which will have the effect of blurring the scattered pixels together within each cluster. The kernel width should be wide enough that points in each cluster merge into a single bump, but not so wide that multiple clusters blur together. Each cluster now corresponds to a blurry spot in the image. Find cluster centers as the local extrema or centers-of-mass of these spots. For example, this could be done using image processing functions to find connected groups of non-white pixels (each group corresponds to a cluster). Find the center of each cluster as the blackest pixel, or take the average pixel location weighted by blackness. This method is essentially performing density-based clustering on a kernel density estimator of black pixel locations.
Approach 2: Threshold the image such that only the black pixels remain on a white background. Extract the xy location of each black pixel. The $(x,y)$ pair for each black pixel is a point in 2d space. Cluster these points using a standard statistical clustering method like k-means, Gaussian mixture models, DBSCAN, etc. You may have to select hyperparameters of the algorithm to obtain a good clustering. For example, some methods require you to specify the number of clusters (there are also automated procedures for choosing this; search this site for details). Cluster centroids are returned by the clustering algorithm (or can otherwise be computed from the points assigned to each cluster).
