* Locals say Internet cut off, supply lines break down
* Businesses shut after deadly unrest over fuel price hike
* Mnangagwa under pressure to fulfil promises on economy
* Govt workers to get supplementary allowance up to March
* Echo of Mugabe-era repression in response to protests (Adds economic fall-out from disorder, Amnesty comment)
By MacDonald Dzirutwe
HARARE, Jan 15 (Reuters) - Zimbabwe's military put on a show of force to deter further unrest on Tuesday after at least three deaths in violent protests over steep fuel price hikes, but continuing disorder blocked the flow of key supplies into the country.
Zimbabweans accuse President Emmerson Mnangagwa of failing to live up to pre-election pledges to kick-start growth, having seen their purchasing power eroded by hyper-inflation, and of resorting to armed forces to crush dissent like strongman predecessor Robert Mugabe.
Mnangagwa has promised a clean break from the 37-year era of Mugabe, who was forced out in a de facto coup in November 2017. But residents said the army was beating up suspected protesters in various townships.
"We are suffering. Mnangagwa has failed this country. Enough is enough, we no longer want this," protester Takura Gomba said in Warren Park, a Harare township, while retreating with others as soldiers approached in trucks.
Amnesty International condemned the military crackdown, saying at least 200 people had been arbitrarily detained and calling on Zimbabwean authorities to ensure restraint by security forces and respect the public's right to protest.
Monday's street disturbances followed sharp increases in fuel prices decreed by Mnangagwa, five months after post-election violence during which six people died when the army intervened to quell trouble.
As security forces faced accusations of heavy-handedness and more protests threatened to break out, Labour Minister Sekai Nzenza announced that public workers would get a monthly supplement of between 5 and 23 percent of their salaries from January to March while wage negotiations with unions continued.
Police spokeswoman Charity Charamba told reporters that a police officer was stoned to death by protesters in Zimbabwe's second city Bulawayo on Monday, while two other people died during protests in Chitungwiza, south of Harare, and Kadoma, west of the capital.
Mnangagwa, who is on an official visit in Moscow, said Zimbabwe was interested in receiving Russian loans and might need Moscow's help in modernising its army, RIA news agency reported.
In Harare and Bulawayo, banks, schools, businesses and the stock market remained shut as many residents stayed at home.
A human rights lawyers' group said it had received reports of soldiers and police breaking into homes in townships overnight and assaulting suspected demonstrators.
Security forces deployed in population centres to stave off further demonstrations, witnesses said, and people in Harare said they could no longer access the internet.
Amnesty, in a statement, said the internet was cut off to prevent people from supporting or organizing protests while critics said the government sought to prevent images of its heavyhandedness from being broadcast around the world.
Information Minister Monica Mutsvangwa said she was not aware of an internet shutdown.
Reports of violent disorder prompted southern African bus lines to suspend services into Zimbabwe, interrupting the flow of goods into Zimbabwe from the diaspora in adjacent countries.
"We can't proceed because they are saying things are tough there. Buses are being burned. Roads are blocked. It's a total shutdown," said Alexio Chirisa. He had been driving a bus full of passengers from the South African city of Port Elizabeth to Harare but was forced to park in Johannesburg on Tuesday.
With Zimbabwe's banks strapped for cash, shipping essential daily commodities like rice and cooking oil to families from their relatives abroad has become a favoured alternative to cash remittances.
"Years of political and economic mismanagement has brought Zimbabwe’s economy to its knees. Millions of Zimbabweans are terrified about the knock-on effects that the fuel increases will have on their daily lives, including for food and healthcare," said Muleya Mwananyanda, Amnesty International's Deputy Director for Southern Africa. (Reporting by MacDonald Dzirutwe in Harare with additional reporting by Joe Brock and Joe Bavier in Johannesburg Editing by Mark Heinrich)