Covariate-adjusted response-adaptive randomization (CARA) designs are gaining increasing attention. These designs combine the advantages of randomized experiments with the ability to adaptively revise treatment allocations based on data collected across multiple stages, enhancing estimation efficiency. Yet, CARA designs often assume that primary outcomes are immediately observable, which is not the case in many clinical scenarios where there is a delay in observing primary outcomes. This assumption can lead to significant missingness and inefficient estimation of treatment effects. To tackle this practical challenge, we propose a CARA experimental strategy integrating delayed primary outcomes with immediately observed surrogate outcomes. Surrogate outcomes are intermediate clinical outcomes that are predictive or correlated with the primary outcome of interest. Our design goal is to improve the estimation efficiency of the average treatment effect (ATE) of the primary outcome utilizing surrogate outcomes. From a methodological perspective, our approach offers two benefits: First, we accommodate arm and covariates-dependent delay mechanisms without imposing any parametric modeling assumptions on the distribution of outcomes. Second, when primary outcomes are not fully observed, surrogate outcomes can guide the adaptive treatment allocation rule. From a theoretical standpoint, we prove the semiparametric efficiency bound of estimating ATE under delayed primary outcomes while incorporating surrogate outcomes. We show that the ATE estimator under our proposed design strategy attains this semiparametric efficiency bound and achieves asymptotic normality. Through theoretical investigations and a synthetic HIV study, we show that our design is more efficient than the design without incorporating any surrogate information.