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Circular photocurrents (CPC), namely circular photogalvanic (CPGE) and photon drag effects, have recently been reported both in monolayer and multilayer transition metal dichalcogenide (TMD) phototransistors. However, the underlying physics for the emergence of these effects are not yet fully understood. In particular, the emergence of CPGE is not compatible with the D3h crystal symmetry of two-dimensional TMDs, and should only be possible if the symmetry of the electronic states is reduced by influences such as an external electric field or mechanical strain. Schottky contacts, nearly ubiquitous in TMD-based transistors, can provide the high electric fields causing a symmetry breaking in the devices. Here, we investigate the effect of these Schottky contacts on the CPC by characterizing the helicity-dependent photoresponse of monolayer MoSe2 devices both with direct metal-MoSe2 Schottky contacts and with h-BN tunnel barriers at the contacts. We find that, when Schottky barriers are present in the device, additional contributions to CPC become allowed, resulting in emergence of CPC for illumination at normal incidence.