Timely sampling and fresh information delivery are important in 6G communications. This is achieved by encoding samples into short packets/codewords for transmission, with potential decoding errors. We consider a broadcasting base station (BS) that samples information from multiple sources and transmits to respective destinations/users, using short-blocklength cyclic and deep learning (DL) based codes for error correction, and cyclic-redundancy-check (CRC) codes for error detection. We use a metric called reported age of information (AoI), abbreviated as RAoI, to measure the freshness of information, which increases from an initial value if the CRC reports a failure, else is reset. We minimize long-term average expected RAoI, subject to constraints on transmission power and distortion, for which we obtain age-agnostic randomized and age-aware drift-plus-penalty policies that decide which user to transmit to, with what message-word length and transmit power, and derive bounds on their performance. Simulations show that longer CRC codes lead to higher RAoI, but the RAoI achieved is closer to the true, genie-aided AoI. DL-based codes achieve lower RAoI. Finally, we conclude that prior AoI optimization literature with finite blocklengths substantially underestimates AoI because they assume that all errors can be detected perfectly without using CRC.