Yes, there is a correlation, and it is negative. Republicans donate more than Democrats.
This is only true if you also count RSIs (Religious Self-identified Institutions). If you exclude those, the numbers no longer favor the republicans.
One problem is that religious recipients for the most part spend the money on themselves.
The latest figures I could find was that the average charity in the US spends around 75% on actual charity, and around 25% for administration and other local costs. However, and this is data from a religious organization, religious charities spend less than 25% on actual charity, and use more than 75% for administration and other local costs.
Someone paying tithe to their church, which is mostly used to pay pastors and preachers and pay for church buildings and whatnot does not help others much. But it seriously skews the figures.
But if you don't believe me, look at the pro capita spending for different countries. There is a direct and strong correlation between the spending pro capita and the ratio of self-identified godless to the overall population. Countries like the Scandinavian countries give the most per person - far more than the US god-botherers give even if counting religious contributions - and also have the lowest ratio of religious citizens.