When can you simplify the $W$ boson propagator? I have seen in several sources that the propagator of the $W$ boson is: 
$$\frac{- i \left( g^{\mu\nu} - \frac{P^\mu P^\nu}{m_W^2} \right)}{p^2 - m_W^2} . $$
But then in some calculations (usually approximations) I have seen that only $$ \frac{-i}{p^2 - m_W^2}   .$$
is used as the propagator. So my question is why is this allowed and  when is it allowed to do this?
 A: You have to be careful, there are many subtleties on the quantization of a spontaneously broken gauge theory like the Standard Model (SM) electroweak sector. I recommend you to learn about the quantization of a gauge theory via path integral methods.  To quantize a gauge theory, you need to introduce a gauge fixing term (and the associated ghost Lagrangian).  For the SM, people usually consider a generic $R_\xi$ guage fixing term.  You can find Feynman rules for the SM in a generic $R_\xi$ gauge fixing in  this article 
https://arxiv.org/abs/1209.6213
In particular look into Eq. 52 for the W-boson propagator.  $\xi_W$ in this Eq. is a free real parameter that comes from the gauge fixing.  When you do calculations, you are free to specify the gauge fixing terms, for instance $\xi_W=1$ corresponds to Feynman Gauge while $\xi_W=0$ to the Landau gauge.  Note that in the Feynman gauge you get rid of the $k_\mu k_\nu$ part. 
As long as you do your calculations consistently, you are assured that the amplitudes for physical processes will always be gauge independent, so you can choose the gauge you prefer.   
