The arbitrarily varying channel (AVC) is a conservative way of modeling an unknown interference, and the corresponding capacity results are pessimistic. We reconsider the Gaussian AVC by relaxing the classical model and thereby weakening the adversarial nature of the interference. We examine three different relaxations. First, we show how a very small amount of common randomness between transmitter and receiver is sufficient to achieve the rates of fully randomized codes. Second, akin to the dirty paper coding problem, we study the impact of an additional interference known to the transmitter. We provide partial capacity results that differ significantly from the standard AVC. Third, we revisit a Gaussian MIMO AVC in which the interference is arbitrary but of limited dimension.