  We give a simple formula for the operator C_3 of the standard deformation
quantization with separation of variables on a K\"ahler manifold M. Unlike C_1
and C_2, this operator can not be expressed in terms of the K\"ahler-Poisson
tensor on M. We modify C_3 to obtain a covariant deformation quantization with
separation of variables up to the third order which is expressed in terms of
the Poisson tensor on M and thus can be defined on an arbitrary complex
manifold endowed with a Poisson bivector field of type (1,1).
