The most devastating cyclone to hit France's poorest department in 90 years caused colossal damage in mid-December, killing at least 39 people and injuring more than 5,600.
Authorities have warned the death toll could rise in the archipelago.
The people of Mayotte "often have the sentiment that what we bring them are assurances, pretty words of solidarity", Bayrou said after visiting a desalination plant.
But what they want is "concrete" action, he said.
After touring areas damaged by the cyclone and meeting local leaders, Bayrou presented dozens of measures in the outlines of a recovery plan that he dubbed "Mayotte standing".
One top priority was for electricity to be "restored to every home by the end of January," he said, promising that state-owned electricity provider EDF would send additional staff and 200 generators.
Two weeks after Cyclone Chido, emergency services are still trying to restore water, power and telecoms services.
"We're not so big here, Mayotte is 374 square kilometres (144 square miles)," said Ali Mohamed, a worker at the archipelago's main hospital.
"It should have been sorted out in two weeks. We've been abandoned, we can see it."
Bayrou had earlier vowed a "second phase" to the recovery plan that would aim to rebuild the archipelago within two years.
"It's not just about rebuilding Mayotte as it was. It's about designing a different future for Mayotte," he added.
Ministers will approve an emergency draft law on reconstruction as early as Friday before its presentation to parliament, Bayrou said on Monday evening, with a longer-term plan set to come before MPs "within three months".
- Slums -
Estelle Youssouffa, a centrist MP representing Mayotte, said France's aid plan was insufficient to meet the needs of the territory's real population, which includes tens of thousands of undocumented migrants.
Mayotte's population stands officially at 320,000, but there are an estimated 100,000 to 200,000 more undocumented inhabitants living in shanty towns that were utterly destroyed by the cyclone.
Bayrou said there would be a "comprehensive and precise census of the population" and trailed further restrictions on the birthright citizenship law that applies on most French soil.
Right-wingers have long demanded an exemption for Mayotte, believing it could deter prospective immigrants, especially from The Comoros.
The right-wing mayor of the capital Mamoudzou, who has called for tougher controls on runaway population growth in Mayotte, showed Bayrou the devastated slums covering the hills around the city.
"We can't let people make the same mistakes and hope the outcome will be different," Ambdilwahedou Soumaila told him.
Both Paris and local authorities would work to "forbid and prevent the reconstruction of shanty towns," Bayrou said as he laid out his recovery plan.
The prime minister had arrived with 2.5 tonnes of aid supplies aboard his plane.
He visited a field hospital and a secondary school in a slum that had been looted after the cyclone.
Troops and police would be deployed to guard schools against such raids, Bayrou said.
He was accompanied to Mayotte by a large delegation of officials, including Education Minister Elisabeth Borne and Manuel Valls, the new overseas territories minister.
The 73-year-old Bayrou, only recently appointed prime minister, had faced criticism for chairing a local council meeting in his home city of Pau in southern France while Mayotte grappled with the aftermath of the deadly cyclone.
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