The wiretap channel models secure communication in the presence of an eavesdropper who must be kept ignorant of transmitted messages. In this paper, the arbitrarily varying wiretap channel (AVWC), in which the channel may vary in an unknown and arbitrary manner from channel use to channel use, is considered. For arbitrarily varying channels (AVCs) the capacity might differ depending on whether deterministic or common randomness (CR) assisted codes are used. The AVWC has been studied for both coding strategies and the relation between the corresponding secrecy capacities has been established. However, a characterization of the CR-assisted secrecy capacity itself or even a general CR-assisted achievable secrecy rate remain open in general for weak and strong secrecy. Here, the secrecy measure of high decoding error at the eavesdropper is considered, where the eavesdropper is further assumed to know channel states and to adapt its decoding strategy accordingly. For this secrecy measure a general CR-assisted achievable secrecy rate is established. The relation between secrecy capacities for different secrecy measures is discussed: The weak and strong secrecy capacities are smaller than or equal to the one for high decoding error. It is conjectured that this relation can be strict for certain channels.