In a nutshell, the problem lies in a mismatch between data behavior and some (strong) assumptions you are implicitly making.
The strongest of these is that the data are one realization of a second-order stationary process. They clearly are not, as you can tell by comparing the region near (450000, 5075000) in the upper "neck" (which I will call "Region X") to other regions within the map. In all other regions, the neighbors of any data support tend to have comparable values. In Region X, however, there are many wild swings between high and low values over short distances. (This is a manifestation of a strong change in the second-order structure.) These swings will inflate the estimates of semivariance at short lags, causing the automatic variogram estimator to infer that the range is short. This range is unlikely to be any greater than a typical distance between an upper and lower extreme point. It is approximately the radius of the colored dots appearing in the map.
One property of Ordinary Kriging is that when all sample points are beyond the range of the variogram, its estimate will be the neighborhood mean. Evidently you have used a fairly large neighborhood--perhaps the entire dataset. This results in many of the kriged values equaling the dataset mean, shown in cyan. The kriged values will vary from this only at grid cells within the range of a data point.
In addition, somehow the automatic fit included almost no nugget effect. This probably was due to mixing in the contributions from the rest of the dataset, where spatial variation is locally small and smooth looking. In particular, there are clusters of tightly spaced points (such as those along the middle left boundary) that would steer the variogram down towards zero at short lags. The absence of any appreciable nugget effect forces the kriged surface to pass through the data points.
These two characteristics of the variogram--short range and small or nonexistent nugget--cause the acned, multiple-bull's-eye pattern to emerge, since the kriged values must change smoothly from the cyan background to each data value within very short distances of each data support.
If you were to change the variogram parameters, then you would wind up smoothing over the variation in Region X. This would hide the interesting and possibly important change of data behavior in Region X.
A more sophisticated approach might combine some of the following elements:
Careful, supervised variography, including cross-validation of the variogram and use of directional and mixed variograms.
Partition of the study area into a priori identifiable regions of stationary behavior.
Co-kriging with the other variables.
Universal kriging to incorporate explanatory covariates.
Initial exploratory data analysis to check the validity of all the data, especially those in Region X.
Adopting more complex models of spatial covariation.
With the limited information available, it would be irresponsible to recommend any particular course of action, since it would depend on why you are kriging, what the data represent, and how you intend to interpret the maps.