解：?$(1)$?∵點(diǎn)?$P $?是點(diǎn)?$M$?關(guān)于點(diǎn)?$N$?的?$“$?半距點(diǎn)?$”$?
∴?$PN=\frac {1}{2}MN$?
①如圖①，∵?$MN=6\ \mathrm {cm}$?，點(diǎn)?$P_{1} $?是點(diǎn)?$M$?關(guān)于點(diǎn)?$N$?的?$“$?半距點(diǎn)?$”$?
∴?$P_{1}N=\frac {1}{2}MN=3(\mathrm {cm})$?
∴?$MP_{1}=MN-P_{1}N=3(\mathrm {cm})$?
②如圖②，∵?$MN=6\ \mathrm {cm}$?，點(diǎn)?$P_{2}$?是點(diǎn)?$M$?關(guān)于點(diǎn)?$N$?的?$“$?半距點(diǎn)?$”$?
∴?$P_{2}N=\frac {1}{2}MN=3(\mathrm {cm})$?
∴?$MP_{2}=MN+P_{2}N=9(\mathrm {cm})$?
綜上，?$MP $?的長(zhǎng)為?$3\ \mathrm {cm} $?或?$9\ \mathrm {cm}$?
?$(2)①$?如圖?$①$?，?$G $?是線段?$MP_{1}$?的中點(diǎn)
∴?$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})$?
②如圖②，?$G_{2}$?是線段?$MP_{2}$?的中點(diǎn)
∴?$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})$?
綜上，線段?$GN$?的長(zhǎng)為?$\frac {9}{2}\mathrm {cm} $?或?$\frac {3}{2}\mathrm {cm}.$?
