A reduced clause is the name commonly given to what is more correctly termed a participial clause (or phrase), e.g. 'serving the food' in
The man serving the food is my brother.
as it is tempting to regard the underlined portion simply as an elliptical relative clause, i.e. an abridged form of
The man WHO IS serving the food is my brother.
However, the fact that such phrases are not simply abbreviated versions of relative clauses but modifying structures in their own right can be seen in the case of stative verbs, since e.g.
Anyone believing that kind of nonsense must be mad.
if similarly extended would result in the nonsentence:
*Anyone WHO IS believing that kind of nonsense must be mad.