Want to ensure that I am reading this correctly. In the PHB it specifies that Wrathful Smite is a concentration spell. However after you hit a target it applies the frightened effect (should it fail a WIS saving throw). So my question is, if I hit the target and it fails its Wisdom Saving throw and the target is now frightened. Does the spell still require my concentration? Or rather the secondary effect just happens and doesn't require me to continue concentrating on the spell?
Concentration is for the duration of the spell
The 1 minute duration of the spell is for both holding the spell until you deliver a hit as well as for maintaining the Condition effect of the smite (in this case, Frightened condition.)
This is supported by Jeremy Crawford via Sage Advice
Some smite spells, such as wrathful smite, do have an ongoing effect that requires concentration
You must continue concentrating
Wrathful Smite says:
The next time you hit with a melee weapon attack during this spell's duration, your attack deals an extra 1d6 psychic damage. Additionally, if the target is a creature, it must make a Wisdom saving throw or be frightened of you until the spell ends. As an action, the creature can make a Wisdom check against your spell save DC to steel its resolve and end this spell.
The frightened effect only lasts until the spell ends (duration: Concentration up to 1 min). As soon as you stop concentrating, the spell ends, which means the frightened effect does too.