解：$(1)$∵点$P $是点$M$关于点$N$的$“$半距点$”$
∴$PN=\frac {1}{2}MN$
①如图①，∵$MN=6\ \mathrm {cm}$，点$P_{1} $是点$M$关于点$N$的$“$半距点$”$
∴$P_{1}N=\frac {1}{2}MN=3(\mathrm {cm})$
∴$MP_{1}=MN-P_{1}N=3(\mathrm {cm})$
②如图②，∵$MN=6\ \mathrm {cm}$，点$P_{2}$是点$M$关于点$N$的$“$半距点$”$
∴$P_{2}N=\frac {1}{2}MN=3(\mathrm {cm})$
∴$MP_{2}=MN+P_{2}N=9(\mathrm {cm})$
综上，$MP $的长为$3\ \mathrm {cm} $或$9\ \mathrm {cm}$
$(2)①$如图$①$，$G $是线段$MP_{1}$的中点
∴$MG_{1}=\frac {1}{2}MP_{1}=\frac {3}{2}(\mathrm {cm})$
∴$G_{1}N=MN-MG_{1}=6-\frac {3}{2}=\frac {9}{2}(\mathrm {cm})$
②如图②，$G_{2}$是线段$MP_{2}$的中点
∴$MG_{2}=\frac {1}{2}MP_{2}=\frac {9}{2}(\mathrm {cm})$
∴$G_{2}N=MN-MG_{2}=6-\frac {9}{2}=\frac {3}{2}(\mathrm {cm})$
综上，线段$GN$的长为$\frac {9}{2}\mathrm {cm} $或$\frac {3}{2}\mathrm {cm}.$
