In　this　paper,　we　mainly　discuss　the　constructions　of　some　new　K-g-frames　which　differ　from　the　existing　methods.　Meanwhile,　we　use　the　relation　between　a　positive　operator　and　the　frame　operator　of　a　K-g-frame　to　yield　a　new　K-g-frame.　We　also　obtain　a　necessary　and　sufficient　condition　to　generate　a　new　K-g-frame.　In　addition,　we　correct　some　recent　results　which　were　obtained　by　Huang　and　Leng.　In　the　end,　we　give　an　equivalent　characterization　to　construct　some　new　tight　K-g-frames　by　two　given　g-Bessel　sequences.　Our　results　generalize　and　improve　some　remarkable　results.　©　2020,　Malaysian　Mathematical　Sciences　Society　and　Penerbit　Universiti　Sains　Malaysia.