A connection on a bundle may be understood as a construction that serves to interpolate between a principal bundle and its characteristic class in real cohomology, represented by curvature characteristic forms of the connection.
If one allows (and it makes good sense to allow this) curvature characteristic forms to be represented not necessarily by globally defined forms, but by cocycles in the abelian sheaf cohomology of the truncated de Rham complex then a weaker notion than that of a connection serves the same purpose: that of a pseudo-connection .
Effectively a pseudo-connection on a principal bundle is locally a collection of Lie-algebra valued 1-forms, that do not have to satisfy the cocycle condition familiar from connections.
That makes pseudo-connections rather empty structures. And in a precise sense this is actually the point of them: there is a (non-concrete) Lie groupoid such that Cech cocycles with values in it are those for -principal bundles with pseudo-connections. This groupoid is in fact weakly equivalent to the delooping groupoid that serves to classify just bare smooth -principal bundles
While for ordinary -principal bundles the notion of pseudo-connection can be (and traditionally is) circumvented, it does become crucial for gerbes and principal 2-bundles and more generally in infinity-Chern-Weil theory. For instance an equivariant gerbe with connection is not an equivariant object in the 2-category of gerbes with ordinary connections, but a certain constrained equivariant gerbe with pseudo-connection.
For the moment see infinity-Chern-Weil theory.
The term pseudo-connection for generalized connections on bundle gerbes apparently first appeared in
Not under this name, but being the same concept, they are considered on bundle gerbes in some detail in