Least Squares with Non Zero Mean Noise
Since this is a linear model if you add noise which isn't centered (Non zero mean noise) your estimation will be good up to a bias term.

The easy way to do so is to remove the bias from $ y $ and solve.  

In case you know the mean of the added noise, just remove it from your measurement and your model becomes the classic Linear Least Squares.

If you don't know it, you have to estimate it. You can solve the problem either in 2 steps, namely estimate, remove, solve classic Linear Least Squares, or build a model which estimates the noise and solve the least squares at once.