Reconfigurable intelligent surface (RIS) is a promising technology for establishing spectral- and energy-efficient wireless networks. In this paper, we study RIS-enhanced orthogonal frequency division multiplexing (OFDM) communications, which generalize the existing RIS-driven context focusing only on frequency-flat channels. Firstly, we introduce the delay adjustable metasurface (DAM) relying on varactor diodes. In contrast to existing reflecting elements, each one in DAM is capable of storing and retrieving the impinging electromagnetic waves upon dynamically controlling its electromagnetically induced transparency (EIT) properties, thus additionally imposing an extra delay onto the reflected incident signals. Secondly, we formulate the rate-maximization problem by jointly optimizing the transmit power allocation and the RIS reflection coefficients as well as the RIS delays. Furthermore, to address the coupling among optimization variables, we propose an efficient algorithm to achieve a high-quality solution for the formulated non-convex design problem by alternately optimizing the transmit power allocation and the RIS reflection pattern, including the reflection coefficients and the delays. Thirdly, to circumvent the high complexity for optimizing the RIS reflection coefficients, we conceive a low-complexity scheme upon aligning the strongest taps of all reflected channels, while ensuring that the maximum delay spread after introducing extra RIS delays does not exceed the length of the cyclic prefix (CP). Finally, simulation results demonstrate that the proposed design significantly improves the OFDM rate performance as well as the RIS's adaptability to wideband signals compared to baseline schemes without employing DAM.