Yes. It is somewhat in between a smile and a grin.
smirk
Smiling, beaming
Wipe that smirk off your face! Please don't smirk at me.
(A smirk may be arrogant, derisive, or innocently playful.)"When the investigators asked him about the crime, the suspect gave a smirk in reply.""I swore that I would win and wipe that arrogant smirk off his face.""Seeing an impish smirk on her son's face, Mrs. Smith suspected he was up to mischief again."
To smile in an affected, often offensively self-satisfied manner
The word smirk is a noun meaning sly smile. Synonyms for the word smirk include beam, grin, leer, simper, grimace, grim, sneer, visage, and smug look.
Grin.
smirk
Josh had a smirk on his face when he got a B+ on his test.
The suffix "-y" can be added to "smirk" to create the word "smirky." This suffix is used to describe someone who often wears a smirk on their face.
Grin or smirk
smirk
perhaps "grimace"
smilestilestalestarestarksharkshirksmirk
"Johnny had a mischievous smirk upon his face after having read his sister's diary."
Smiling, beaming
quirk, smirk, shirk