Yes plants can absorb organic compounds..For example plants can absorb citric acid
Due to citric acid’s ability to chelate metals and be absorbed by plants, it is hypothe-sized that exposure to it will increase the ability of wheat-grass to absorb macro- and micro-nutrients, as well as heavy metals, from soil.
Citric acid chelates absorption
Also plants have the ability to absorb and release sugars.
Plant roots are able to absorb sugars from the rhizosphere but also release sugars and other metabolites that are critical for growth and environmental signaling. Reabsorption of released sugar molecules could help reduce the loss of photosynthetically fixed carbon through the roots
Monosacharid absorption activity
I'm sure there are other examples, feel free to edit the answer.