In my answer I will explain how each point, argument and source etc. affects how Edwards influence upon the Normans reached or maintained a high or low point within this time period. I will also say how these pieces of information are for and against the Influence towards the Normans.
We firstly know that Edward Was born in England but raised in Normandy thus meaning that Normandy could have first acquired a taste upon him meaning that he would prefer it to England. At this time there were many Viking raids going on within England. Sweyn took the throne in 1013 but later died in 1014 thus Aethlred sent Edward back to his ambassadors. Aethlred died in 1016 making Edmund Ironside (Edwards’s half brother) take up the fight against Cnut (Sweyn’s son) He also died leaving Edward to go into exile with his brother and sister, but his mother married Cnut even after all he had done possibly making an even bigger hatred for anything or anyone to do with England.
This is when Edward spent quarter of a century in exile within Normandy helping charters within Normandy signing them as potential kings of England. The man’s names were William of Jumièges, Robert I, Duke of Normandy. He also received support in Normandy during this time from his sister, Godgifu, who was married to Drogo of Mantes, count of Vexin. These two Normans along with Edward invaded England to get Edward on the throne but this did not work. He also received support from a small amount of continental abbots in particular Robert, abbot of the Norman abbey of Jumieges. He would later become Edwards Archbishop of Canterbury.
There is little amount of Norman influence during Edwards return to England. Harthacnuts death on the 8th of June, 1042 meant that Edward could succeed the throne but would need help which he received from the likes of Earl Godwin of Wessex. He also received of popularity from the people within London even