  The aim of this paper is to develop a refined error estimate of L1/finite
element scheme for a reaction-subdiffusion equation with constant delay $\tau$
and uniform time mesh. Under the non-uniform multi-singularity assumption of
exact solution in time, the local truncation errors of the L1 scheme with
uniform mesh is investigated. Then we introduce a fully discrete finite element
scheme of the considered problem. Next, a novel discrete fractional Gr\"onwall
inequality with constant delay term is proposed, which does not include the
increasing Mittag-Leffler function comparing with some popular other cases. By
applying this Gr\"onwall inequality, we obtain the pointwise-in-time and
piecewise-in-time error estimates of the finite element scheme without the
Mittag-Leffler function. In particular, the latter shows that, for the
considered interval $((i-1)\tau,i\tau]$, although the convergence in time is
low for $i=1$, it will be improved as the increasing $i$, which is consistent
with the factual assumption that the smoothness of the solution will be
improved as the increasing $i$. Finally, we present some numerical tests to
verify the developed theory.
