Is it necessary to have a Cross with a corpus (an image of the body of Christ) at the entrance procession of the Mass?

Yes. The General Instruction of the Roman Missal 2010 changed what had become a custom, namely to use a cross with the image of the resurrected Christ. Now the GIRM #117 states: "...there is to be a cross adorned with a figure of Christ crucified." Such a cross may be carried in procession and then placed near the altar…”in which case it ought to be the only cross used...”  If there is a crucifix in the worship space the processional cross should be placed in a dignified place away from the altar. We are reminded of the sacrificial aspect of the Eucharist, the “unbloody sacrifice of Calvary” by the presence of this crucifix.