Puzzled by CIC filters
It seems you mixed up CIC filters and recursive moving-average

A recursive moving-average uses a comb-filter and an accumulator.

$$ H(z) = \frac{1}{N}\frac{1-z^{-N}}{1-z^{-1}} $$

The numerator corresponds to a comb filter (i.e a delay line of N elements and 1 adder).
The denominator corresponds to an accumulator. The 1/N factor is to get a moving-average, otherwise you'd have a moving-accumulator. You only need 2 additions, a delay line, an accumulator and a divider. With a non-recursive moving-average filter, you'd need N additions and you'd need a delay line anyway. A recursive moving-average is much 
less computionally expansive than a non-recursive moving-average filter.

Now the CIC filter combines the recursive moving-average filter with decimation and interpolation.

For a CIC decimation filter, you insert the rate transition block between the accumulator and the comb filter. Say you have a 16-sample moving average-filter (N=16) and you want to decimate by 8 (R = 8). You'd need a delay line of 16 elements. However, by inserting the decimation between the accumulator and the comb filter, you can reduce the number of elements in the delay line. You'd only need N/R elements, i.e. 2. You'd still need to subtract 2 numbers, however you would do it at sampling rate of fs/8.

I suggest you consult this site

https://www.researchgate.net/publication/228648728_Understanding_cascaded_integrator-comb_filters

Update: see this nice write-up by Richard Lyons-
https://www.dsprelated.com/showarticle/1337.php