An exact analysis of a three-state Potts model with infinite-range interactions in the presence of an asymmetric random field is carried out. Complete phase diagrams at zero temperature are obtained. By finding lines of critical points and lines of tricritical points, the asymmetrical random field is shown to have strong effects on phase transitions. Under certain circumstances, a sufficient asymmetry in the distribution of the random field can either terminate a first-order phase transition at a critical point or bring it to a second-order phase transition through a tricritical point.