Hi. Hand injuries even if small can be extremely painful and if not managed with required physiotherapy may be quite troublesome. Even with pain are you able to flex and extend the finger(bend inside and outside)? I.e how deep was the cut?
Next Steps
If there was no tendon injury then all you need is good physiotherapy and massaging of the scar area and keeping it supple with good moisturiser (even coconut oil serves the purpose) and silicone gels. However, if there is no movement at all then u will have to looked at once..