We introduce the set of $g$-attractive points of a mapping $f$ and study a $g$-averaged Halpern-type iterative scheme for $b$-enriched $g$-nonexpansive mappings in real Hilbert spaces. For $0<\lambda \le 1/(b+1)$, we prove that every sequence generated by the scheme converges strongly-after applying $g$-to the metric projection of the anchor onto the set $A^g(f_{g,\lambda })$, provided this set is nonempty. As a consequence, we obtain the corresponding strong convergence theorem for $b$-enriched nonexpansive mappings. A numerical example is given to illustrate the method and the role of the averaging parameter.