Pseudo core invertibility in a certain finite-dimensional $\ast$-algebra generated by two projections

We investigate the existence criterion and formulae of pseudo core inverses of elements in a certain finite-dimensional $\ast$-algebra generated by two projections $p$ and $q$ under some prescribed conditions that $(pq)^2=pq$ and $pqp=\lambda p (\lambda \in \mathbb{F})$, respectively. In addition, we solve a question about Drazin invertibility proposed by Chen et al.