A graph is claw-free if it contains no induced subgraph isomorphic to a K_1,3. This paper studies hamiltonicity in two subclasses of claw-free graphs. A claw-free graph is CN-free (claw-free, net-free) if it does not contain an induced subgraph isomorphic to a net (a triangle with a pendant leaf dangling from each vertex). We give a structural characterisation of CN-free graphs which yields a simple proof that any connected (1-tough) CN-free graph has a Hamilton path (cycle). We strengthen this result for CN-free graphs of higher connectivity. For example, we show 3-connected CN-free graphs are both Hamilton connected and pancyclic. In addition, the characterisation yields a polynomial algorithm for finding a Hamilton path (cycle) in a connected (2-connected) CN-free graph. The second class consists of those claw-free graphs such that for each vertex v the set of vertices at distance two from v does not contain an independent subset of size three. We call such graphs distance claw-free and give a forbidden subgraph characterisation. We use this along with our earlier characterisation to prove that any 2-connected (3/2-tough) distance claw-free graph has a Hamilton path (cycle). Our characterisation also gives a polynomial time algorithm to solve the weighted stable set problem in a distance claw-free graph.