Smoothing a power spectrum
Two things: 1) Unless there's some kind of synchronicity between your spectrum analyzer oscillator and the signal source, the spectrum should be fairly symmetrical around 0Hz (except for the AWGN). So I strongly suspect something wrong there, because it isn't symmetrical. 2) Since (I assume) you are getting more than one spectral sweep, for each frequency bin take something like the average of the bin over multiple sweeps. For example, a given frequency bin could be the average of the last several sweeps for that bin, or its an IIR of FIR of the bin history. Do NOT smooth across frequency unless you only have one sweep of data and nothing else.