Correlation between genome size and mutation rate?
Bradwell et al. report RNA viruses with the smallest genome may have particularly high mutation rates. A particular bug was shown to have about 1.4 x $10^{-4}$ substitutions per nucleotide per round of copying (high for viruses generally) and the authors suggest this is to optimize adaptibility. Correlation Between Mutation Rate and Genome Size in Riboviruses: Mutation Rate of Bacteriophage $Q \beta$, Genetics, vol.195 no. 1 243-51 (2013). 

Cuevas et al. report a phage with 1.0 x $10^{-6}$ mutations per base per round of copying (m/b/r), which conforms (they state) to Drake's rule of 0.003 mutations per genome per round of copying in DNA-based microorganisms. They indicate that DNA-based microorganisms range between $10^{-10}$ to $10^{-6}$ m/b/r. They state that RNA viruses vary between $10^{-6}$ and $10^{-4}$ m/b/r, consistent with the result above. Point Mutation Rate of Bacteriophage $\Phi X174,$ Genetics, vol. 183 no 2 (2009).

Drake's paper often-cited paper (one of them) is available at this link. 
He has a remarkable graph on p. 7163 of this paper in which based on his observation of seven microorganisms with a wide range of genome size (bp) he plots the log of the mutation rate per base pair against the log of genome size (bp). The resulting (fitted) line has a slope of roughly negative one. 

In a nutshell, Drake shows that over a wide range of genome size, as the genome size climbs exponentially the mutations per base per round of copying drops exponentially. This relationship is roughly log-log linear. 

In his discussion he says that the factors influencing mutation rates (in either direction) are so general that they seem to have imposed a small range of variation of mutation rate per genome in microbes that have a large range in genome size and mutation rate per base pair.   

As for more complex organisms, I have read very little. There is a recent review article in Nature about a report that a hypothesized inverse correlation between genome size and mutation rates in certain plants (angiosperms) is untrue. Genome size is positively correlated with genome size. Plants with the larger genomes have higher mutation rates than those with smaller than average genomes. I don't have access to this pay-walled article or the methods involved so I am quoting the abstract. Tasci, Nature Reviews Genetics 13, 148 (March 2012).     

As usual an expert in this area could correct/expand these few details I've found but it's an interesting question. Drake's paper gives a remarkable result. 

As for the question about correlation coefficient, I think that Drake's graph suggests a strong correlation between linearized functions over a certain range. 
I suppose we could attach a number to that...