Chirality is not well-defined for massive fields. A famous consequence of this fact are pion masses, which can be linked to Chiral Symmetry Breaking.
In the Lagrangian, you can define left- and right-handed Weyl fermions independently. A mass term will mix these, giving a massive Dirac fermion. Weyl fermions fulfill either $$ P_{L} \psi_L = \psi_L, \quad \text{or} \quad P_R \psi_R = \psi_R$$ but a Dirac fermion is not an eigenstate of the projection operators $$ P_{L,R} \psi_D \neq \alpha \psi_D. $$
There is a computational trick called a "mass insertion", which can be confusing in this regard:
A Dirac fermion can be considered as a coupled system of two Weyl fermions, where the mass is the coupling parameter. If a fermion's mass is small compared to the energy of a given process, one can approximate the Dirac fermion by its two (massless) Weyl components. The advantage is that for massless fields, loop integrals usually take much simpler forms.
Corrections to the massless case can then be included by adding a Feynman rule for the mass term in the Lagrangian, which is a bilinear coupling between the left- and right-handed Weyl fermions. If you were to resum all possible mass insertions, the result is the same as if you had started with the massive Dirac fermion from the start. Since the underlying assumption of the approximation is that the mass is small compared to other energy scales in the theory, the corrections are usually small, though.
Sometimes, the diagram including a mass insertion is computed in order to show that the error induced by neglecting the mass is small indeed.