We study history-dependent online problems with a possibly inaccurate prediction of the future request sequence. Motivated by several real-world applications, we introduce a \emph{bounded-influence} framework in which past decisions and requests affect the future optimal value by only a bounded amount. Within this framework, we develop AdaSwitch, a meta-algorithm that adaptively switches between suitable offline and online oracles. AdaSwitch provides explicit guarantees on expected performance that tighten as prediction error decreases or the offline optimum increases. With perfect predictions, its guarantee approaches the offline oracle's guarantee as the offline optimum grows. It also retains a worst-case guarantee close to that of the online oracle under arbitrary predictions. Applications to online lead-time quotation, $k$-server and caching, and online reusable resource allocation demonstrate the framework's applicability to both reward maximization and cost minimization.

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