On of the major effects that is often under appreciated is the 'liming ' effect of ash and biochar. Unless you have a 'control' treatment that compensates for this effect you will not get any study published in a reputable journal.
The Mn is taken up as the reduced +2 species which is less stable in aerobic systems the higher the pH unless coordinated. Oxidation is microbial and the rate increases with pH. Consequently, the combination of high pH and DOC suggests to me a low Mn bioavailability.
Yes Mn is absorbed by roots as Mn2+ so the uptake is depending on the availability of the free ion Mn2+, which can be affected by the DOC. High DOC may reduce Mn2+ concentration by making complex with the free ion but on the other hand the DOC-Mn complex can dissociate at the root surface and buffer the availability of the free ion. Considering these two opposite mechanisms, it is difficult conclude, it depends additionally on the sorption onto the solid phase and on te lability of the complex. Modelling should be the solution, both speciation modelling to see the effect of DOC on the Mn2+ concentration and transport-reaction model to finally assess how DOC-Mn complex buffer the availability of uptake. Here is a paper that can help :
Lin, Z., Schneider, A., Sterckeman, T., Nguyen, C., 2015. Ranking of mechanisms governing the phytoavailability of cadmium in agricultural soils using a mechanistic model. Plant and Soil 1–19. doi:10.1007/s11104-015-2663-6