  We present a novel construction of finite groupoids whose Cayley graphs have
large girth even w.r.t. a discounted distance measure that contracts
arbitrarily long sequences of edges from the same colour class (sub-groupoid),
and only counts transitions between colour classes (cosets). These groupoids
are employed towards a generic construction method for finite hypergraphs that
realise specified overlap patterns and avoid small cyclic configurations. The
constructions are based on reduced products with groupoids generated by the
elementary local extension steps, and can be made to preserve the symmetries of
the given overlap pattern. In particular, we obtain highly symmetric, finite
hypergraph coverings without short cycles. The groupoids and their application
in reduced products are sufficiently generic to be applicable to other
constructions that are specified in terms of local glueing operations and
require global finite closure.
