  Previous works have considered the leading correction term to the scaled
limit of various correlation functions and distributions for classical random
matrix ensembles and their $\beta$ generalisations at the hard and soft edge.
It has been found that the functional form of this correction is given by a
derivative operation applied to the leading term. In the present work we
compute the leading correction term of the correlation kernel at the spectrum
singularity for the circular Jacobi ensemble with Dyson indices $\beta = 1,2$
and 4, and also to the spectral density in the corresponding $\beta$-ensemble
with $\beta$ even. The former requires an analysis involving the
Routh-Romanovski polynomials, while the latter is based on multidimensional
integral formulas for generalised hypergeometric series based on Jack
polynomials. In all cases this correction term is found to be related to the
leading term by a derivative operation.
