The authors of Fluxes and sources of volatiles discharged from Kudryavy, a subduction zone volcano, Kurile Islands write:
Several potential sources for CO2 can be identified: (1) hotspot-type mantle, (2) MORB-type mantle, (3) subducted oceanic crust (MORB), (4) marine carbonate, (5) organic C from crustal and subducted sediments
They also give a table which states that in island arc volcanoes, 12% of carbon is sourced from the mantle, 67% is sourced from carbonates and 21% is organic carbon. Note that this only gives the amount degassed from volcanoes in subduction zone arcs. This does not mean that all carbonate-hosted carbon is released from the rocks to be degassed. I read somewhere that only 50% is degassed, and the rest is lost to the deep mantle. I would also expect that MORB magmas will have significantly less carbonate-sourced carbon.
Also, The deep carbon cycle and melting in Earth's interior may be of interest.