Abstract: This paper studies two fundamental problems in regularized Graphon Mean-Field Games (GMFGs). First, we establish the existence of a Nash Equilibrium (NE) of any $\lambda$-regularized GMFG (for $\lambda\geq 0$). This result relies on weaker conditions than previous works analyzing both unregularized GMFGs ($\lambda=0$) and $\lambda$-regularized MFGs, which are special cases of GMFGs. Second, we propose provably efficient algorithms to learn the NE in weakly monotone GMFGs, motivated by Lasry and Lions (2007). Previous literature either only analyzed continuous-time algorithms or required extra conditions to analyze discrete-time algorithms. In contrast, we design a discrete-time algorithm and derive its convergence rate solely under weakly monotone conditions. Furthermore, we develop and analyze the action-value function estimation procedure during the online learning process, which is absent from algorithms for monotone GMFGs. This serves as a sub-module in our optimization algorithm. The efficiency of the designed algorithm is corroborated by empirical evaluations.