Given $H(x)$ is the Heaviside Theta, the tables give the following Fourier transforms for it:

$$ H(x+a)\to -PV\frac{i e^{i a w} }{w}+\pi \delta (w)$$

while from Sokhotski–Plemelj theorem it follows (seemingly) that

$$ H(x+a)\to~PV\frac{-i}{w} +\pi\delta(w)+ |a|e^{iax/2}\operatorname {sinc} \left({\frac {w a }{2\pi}}\right)$$

The first result seems questionable to me because it does not depend on $a$ at $w=0$. But this should not be the case because $H(x+a)$ can be represented as a sum of $H(x)$ and the $rect$ function, and the Fourier transform of the later depends on $a$.