Karen Mead read a poem written by the couple's nine-year-old daughter Darcie and asked for anyone who could help in the search to come forward.
Mrs Mead, of High Green, Sheffield, last saw her husband on Saturday, July 31, when he left their home. It is thought he may have spent a night in a city hotel.
The 36-year-old's car was found abandoned on Tuesday, August 3 near the Ladybower reservoir in the Peak District, an area where he is known to go walking.
Mrs Mead said: "We all love you and we all want you back including the children. Any problems can be dealt with. We all love and miss you. You have a great future."
She then read the poem which said: "I love you so much I think the world of you. I really want you to come back home. When you come back, I will give you the biggest hug."
Supt Martin Scothern, of South Yorkshire Police, said: "Despite searches to locate Jason during the past three weeks, including appeals in the media, his disappearance remains unexplained.
"We are following all lines of inquiry and are keeping an open mind at this stage. We need people to come forward with any information to help us find him however trivial or irrelevant they think it is. We remain hopeful that Jason will return safely."
Police said the last known communication from Mr Mead was a text message sent on August 1 from his personal mobile, which was found in the car. A Facebook page has been set up in an attempt to help in the search.
Anyone with information about Mr Mead is urged to call South Yorkshire Police on 0114 220 2020 quoting incident number 260 of August 8, 2010.