Different from traditional knowledge graphs (KGs) where facts are represented as entity-relation-entity triplets, hyper-relational KGs (HKGs) allow triplets to be associated with additional relation-entity pairs (a.k.a qualifiers) to convey more complex information. How to effectively and efficiently model the triplet-qualifier relationship for prediction tasks such as HKG completion is an open challenge for research. This paper proposes to improve the best-performing method in HKG completion, namely STARE, by introducing two novel revisions: (1) Replacing the computation-heavy graph neural network module with light-weight entity/relation embedding processing techniques for efficiency improvement without sacrificing effectiveness; (2) Adding a qualifier-oriented auxiliary training task for boosting the prediction power of our approach on HKG completion. The proposed approach consistently outperforms STARE in our experiments on three benchmark datasets, with significantly improved computational efficiency.