In this paper, we consider a new multi-armed bandit (MAB) framework motivated by three common complications in online recommender systems in practice: (i) the platform (learning agent) cannot sample an intended product directly and has to incentivize customers to select this product (e.g., promotions and coupons); (ii) customer feedbacks are often received later than their selection times; and (iii) customer preferences among products are influenced and reinforced by historical feedbacks. From the platform's perspective, the goal of the MAB framework is to maximize total reward without incurring excessive incentive costs. A major challenge of this MAB framework is that the loss of information caused by feedback delay complicates both user preference evolution and arm incentivizing decisions, both of which are already highly non-trivial even by themselves. Toward this end, we first propose a policy called ``UCB-Filtering-with-Delayed-Feedback'' (UCB-FDF) policy for this new MAB framework. In our analysis, we consider delayed feedbacks that can have either arm-independent or arm-dependent distributions. In both cases, we allow unbounded support for the random delays, i.e., the random delay can be infinite. We show that the delay impacts in both cases can still be upper bounded by an additive penalty on both the regret and total incentive costs. This further implies that logarithmic regret and incentive cost growth rates are achievable under this new MAB framework. Experimental results corroborate our theoretical analysis on both regret and incentive costs.