With　the　rise　of　the　processing　power　of　networked　agents　in　the　last　decade,　second-order　methods　for　machine　learning　have　received　increasing　attention.　To　solve　the　distributed　optimization　problems　over　multiagent　systems,　Newton＇s　method　has　the　benefits　of　fast　convergence　and　high　estimation　accuracy.　In　this　article,　we　propose　a　reinforced　network　Newton　method　with　K-order　control　flexibility　(RNN-K)　in　a　distributed　manner　by　integrating　the　consensus　strategy　and　the　latest　knowledge　across　the　network　into　local　descent　direction.　The　key　component　of　our　method　is　to　make　the　best　of　intermediate　results　from　the　local　neighborhood　to　learn　global　knowledge,　not　just　for　the　consensus　effect　like　most　existing　works,　including　the　gradient　descent　and　Newton　methods　as　well　as　their　refinements.　Such　a　reinforcement　enables　revitalizing　the　traditional　iterative　consensus　strategy　to　accelerate　the　descent　of　the　Newton　direction.　The　biggest　difficulty　to　design　the　approximated　Newton　descent　in　distributed　settings　is　addressed　by　using　a　special　Taylor　expansion　that　follows　the　matrix　splitting　technique.　Based　on　the　truncation　on　the　Taylor　series,　our　method　also　presents　a　tradeoff　effect　between　estimation　accuracy　and　computation/communication　cost,　which　provides　the　control　flexibility　as　a　practical　consideration.　We　derive　theoretically　the　sufficient　conditions　for　the　convergence　of　the　proposed　RNN-K　method　of　at　least　a　linear　rate.　The　simulation　results　illustrate　the　performance　effectiveness　by　being　applied　to　three　types　of　distributed　optimization　problems　that　arise　frequently　in　machine-learning　scenarios.