Pedzisa mutsara unotevera: Mwaka wa2004-05 wakaona shanduko kumutemo uyu, sezvo mubatanidzwa wakatanga kushandisa tafura yeCeltic League seyayo yega nzira yekusarudza kuti ndeapi zvikwata zveIrish zvaizopinda mumwaka unotevera weHeineken Cup. Connacht yakapedza matanhatu matanho shure kweUlster mumakwikwi ekupedzisira gore iroro, ichipotsa zvakare. Kwakanga kusati kuri kutozosvikira
2016 iyo Connacht yakapedza iri panzvimbo yakakwirira yeligi kupfuura chero ipi zvayo yevakwikwidzi vavo zvakare. Kunyangwe izvi Connacht akawana mukana weHeineken Cup kubva mwaka wa2011-12 kusvika mwaka wa2013-14. Nenzira inosetsa ikoku kwakanga kuri nemhaka yaLeinster, uyo akakunda makwikwi matatu anotevedzana eEurope. Kubudirira kweLeinster kwaireva kuti ivo vakakodzera otomatiki kwegore rinotevera, vachisiya nzvimbo yekuIreland yekukodzera yakavhurika iyo yakazadzwa naConnacht.