The road has been closed to all traffic except buses for years now, as motorists were previously accused of using it as a “rat run” to get to Ikea.
Mr Kenny is now calling on Midlothian Council to reverse the car ban.
He said: “It needs to be opened up again. I think the pub there is desperate to get it opened again. Other than those living on that street, people in Loanhead would like to see it open again.
"It’s the footprint for everybody driving further than they should need to if that road was open. It also places extra traffic on a main trunk road at peak times.”
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A Midlothian Council spokesman said: “We responded to community concerns in the road closure and have no plans to reopen Dryden Terrace to vehicles, but we would consult with local people, elected members and the community council if there were any planned changes.”