Pseudodiagrams are knot or link diagrams where some of the crossing information is missing. Pseudoknots are equivalence classes of pseudodiagrams, where equivalence is generated by a natural set of Reidemeister moves. In this paper, we introduce a Gauss-diagrammatic theory for pseudoknots which gives rise to the notion of a virtual pseudoknot. We provide new, easily computable isotopy and homotopy invariants for classical and virtual pseudodiagrams. We also give tables of unknotting numbers for homotopically trivial pseudoknots and homotopy classes of homotopically nontrivial pseudoknots. Since pseudoknots are closely related to singular knots, this work also has implications for the classification of classical and virtual singular knots.