Context: We consider only digraphs. Let CYCLE be the language of graphs with a cycle; it is an NL-complete problem. Let HASEDGE be the language of graphs with at least one edge. Then trivially, $\text{CYCLE} \cup \text{HASEDGE}$ is no longer NL-hard, while $\text{CYCLE} \cup \overline{\text{HASEDGE}}$ stays so.
Actual problem: I'm wondering if the language $$\text{CYCLE} \cup \{(V, E):(\exists u,v,x,y)[E(u, v) \land E(x, y) \land \neg E(u, y) \land \neg E(x, v)]\}$$ is still NL-hard.
Question: For which FO formula $\phi$ on the vocabulary of graphs is $$\text{CYCLE} \cup \{(V, E) : (V, E) \models \phi\}$$ NL-hard? Is this property decidable?
Thanks for your input!