How to automatically identify a surge in a periodic signal
My suggestion would be to plot the frequency response by taking the FFT of a reasonable length to see the spectrum. Apart from the two spikes in FFT output (corresponding to sinewave) you could see the DFT of the periodic spur signal with approximate amplitude of $N$ where $N$ is the period of the combined signal. The DFT of spur is a constant amplitude signal, which means, for all FFT bins, it will be constant value. If your signal is noisy, you could easily average out these values to find the amplitude of spur. You should first subtract amplitude of sinewave from the FFT before computing this. Here is the code I tried

clc
clear all
n=0:9999;
x = sin(pi/5*n);
spike = repmat([1 zeros(1,9)],1,1000);
x1 = x + spike;
plot(abs(fftshift(fft(x1,1000))))


I got the amplitude of spur signal to be $N/N_{fft}$.