Is it always true (because of covering spaces has homotopy lifting property)?
loop $f$ lifts to a closed curve if and only if any curve freely homotopic to $f$ lifts to a closed curve.
or we have to assume that cover is normal? (In that case could anyone present any counterexample and proof of this fact for normal cover)
I ask because I found a Remark:
Since the cover in the conclusion of Lemma 2.1 is normal, a closed curve $f$ lifts to a closed curve if and only if any curve freely homotopic to $f$ lifts to a closed curve.
I found it in the paper: Malestein, Justin, Putman, Andrew; On the self-intersections of curves deep in the lower central series of a surface group. Geom. Dedicata 149 (2010), 73–84. which is available here.