Let $$G$$ G be a $$K_{1,3}$$ K 1 , 3 -free graph. A circuit of $$G$$ G is essential if it contains a non-locally connected vertex $$v$$ v and passes through both components of $$N(v)$$ N ( v ) . The essential girth of $$G$$ G , denoted by $$g_e(G)$$ g e ( G ) , is the length of a shortest essential circuit. It can be seen easily that, by Ryjáček closure operation, the essential girth of $$G$$ G is closely related to the girth of $$H$$ H where $$H$$ H is the Ryjáček closure of $$G$$ G and is a line graph. A generalized net, denoted by $$N_{i_1,i_2,i_3}$$ N i 1 , i 2 , i 3 , is a graph obtained from a triangle $$C_3$$ C 3 and three disjoint paths $$P_{i_\mu +1}$$ P i μ + 1 ( $$\mu =1,2,3$$ μ = 1 , 2 , 3 ), by identifying each vertex $$v_\mu $$ v μ of $$C_3=v_1v_2v_3v_1$$ C 3 = v 1 v 2 v 3 v 1 with an end vertex of the path $$P_{i_\mu +1}$$ P i μ + 1 , for every $$\mu = 1,2,3$$ μ = 1 , 2 , 3 . In this paper, we prove that every $$2$$ 2 -connected $$\{ K_{1,3}, N_{1,1,g_e(G)-4}\}$$ { K 1 , 3 , N 1 , 1 , g e ( G ) - 4 } -free (and $$\{ K_{1,3}, N_{1,0,g_e(G)-3}\}$$ { K 1 , 3 , N 1 , 0 , g e ( G ) - 3 } -free) graph $$G$$ G contains a Hamilton circuit. With the additional parameter $$g_e$$ g e , these results extend some earlier theorems about Hamilton circuits in $$\{ K_{1,3}, N_{a,b,c}\}$$ { K 1 , 3 , N a , b , c } -free graphs (for some small integers $$a, b$$ a , b and $$c$$ c ).