So, it seems that most people think Larkin are some sort of "Investiture Vacuums" that just consume any Investiture around them. However, I think that this isn't the case. When Rsyn gets her Larkin, there isn't really anything different about it (other than the silver eyes and wings), but when Nale uses the Larkin on Lift, it is described as sickly and weak looking. Therefore, I must conclude that Larkin just have the ability to use Stormlight, like a Surgebinder. The Larkin drew in the Stormlight to heal itself from injuries that Nale likely gave it. Naturally, to draw in Stormlight, the Larkin must have a bond with a spren. However, this is probably a higher spren, such as something below Radiantspren but above normal spren. The most likely candidates that we know of are Riverspren and Windspren, though there may be others.
I hope I cleared some things up.