This means fully-vaccinated workers in these sectors will be able to escape self-isolation rules even if they come into contact with someone who tested positive.
It comes after dozens of industry bosses warned of staffing and product shortages caused by the “pingdemic“.
More than 600,000 people in England and Wales were told to quarantine by the NHS Covid-19 app in the week to July 14.
However Richard Walker, managing director of the Iceland chain, said the plan only applied to depot and factory workers “so it only fixes half the supply chain issue”.
Speaking to the BBC, he said there is “plenty of stock to go around” but that his staff are having to work “very hard to keep the show on the road”.
The boss said additional problems include the driver shortage and staff having to isolate.
He said: “There’s a lot of red tape in getting these testing centres set up, it’s a nightmare so with three weeks to go, in my view, it’s simply not worth the effort.”
Mr Walker said he believes staff could potentially start deleting the app “en masse, in numbers that they weren’t previously doing, and to be honest who can blame them”.
On Monday, the government said organisations were being contacted by NHS Test and Trace so they could mobilise sites this week to test up to 10,000 food service staff.
But Birchall Foodservic, which delivers food to schools, hospices and care homes, said it had not received any communication about whether they are included in the scheme.
A Government spokesperson said:“We are working closely with the food sector and have engaged with retailers who have identified that the distribution sides of their businesses are where workplace testing is most practical. This will also ensure food gets to stores on time.
“Over 500 priority test sites have been identified across supermarket and wholesale distribution centres, large manufacturing sites and processing plants – all of these sites have been engaged by Test and Trace. Already several of the major supermarket depots have set up testing sites.”