Convolutional codes are a class of error-correcting codes that performs very well over erasure channels with low delay requirements. In particular, Maximum Distance Profile (MDP) convolutional codes, which are defined to have optimal column distances, are able to correct a maximal number of erasures in decoding windows of fixed sizes. However, the required field size in the known constructions for MDP convolutional codes increases rapidly with the code parameters. On the other hand, if the code parameters are small, larger bursts of erasures cannot be corrected. In this paper, we present a new class of convolutional codes, which we call Pseudo-MDP convolutional codes. By definition these codes can correct large bursts of erasures within a prescribed time-delay and still keep part of the advantageous properties of MDP convolutional codes, in the sense that we require some but not all column distances to be optimal. This release in the condition on the column distances allows us to construct Pseudo-MDP convolutional codes over fields of smaller size than those required for MDP convolutional codes with the same code parameters.