Well one hack would be to generate the needed data for the histogram from your specification. So, let's say your spec what you suggested, e.g. {value,nValues}. Then you could recreate the data from this by using ConstantArray on it. Here's an example:

dataSpec = {{1, 12}, {3, 4}, {9, 34}, {32, 7}};

histogramData = ConstantArray[#[[1]], #[[2]]] & /@ dataSpec

Histogram[histogramData, 40]

The average is just taking each data item times its frequency and adding them all together and then dividing by the total of all the frequencies. So....

In[2]:= dataSpec = {{1, 12}, {3, 4}, {9, 34}, {32, 7}};

In[3]:= data = dataSpec[[All, 1]]

Out[3]= {1, 3, 9, 32}

In[4]:= frequencies = dataSpec[[All, 2]]

Out[4]= {12, 4, 34, 7}

In[6]:= Tr[data frequencies]/Tr[frequencies]

Out[6]= 554/57