He also signalled that he believes the current package of measures are not enough to support British households amid a worsening economic outlook.
Inflation has dominated the race to replace Mr Johnson as frontrunner Liz Truss and Rishi Sunak offer competing visions of the best way to ease the pressure on struggling households.
Mr Johnson, speaking during a visit to north Wales, said that whoever is the next prime minister will focus on delivering extra measures to help those hit by the rising cost of living.
He also said the Government had already announced a series of measures designed to support people, but added that not everyone will yet have received that help.
Earlier, Chancellor Nadhim Zahawi said that he believed the next prime minister can “hit the ground running” to help hard-pressed families.
He said that his officials in the Treasury are “looking at all the options” for additional help this winter to ensure that either Ms Truss or Mr Sunak have the necessary information to take decisions from September 5.
Mr Johnson said he understood that things are not “easy” for many people, but insisted that there will be further help in the new year.
He told reporters: “The price of energy we’re going to bring down by investment in British supply in renewables, in nuclear and all the other.
“I’m not going to pretend that things are easy for people right now. You’re right to push me because we’re doing everything that we can.
“But there’s more money coming anyway, as a result of the decisions being taken, there will be further help coming in October and in the new year.”
He said the Government was taking short and long-term steps to address the energy crisis.
Mr Johnson added: “What we are doing is making sure that we have the fiscal firepower, we have the cash, to help people more at those two key points, in October and January.”
Asked if he believed the current package of support was enough, he said: “No, because what I’m saying what we’re doing in addition is trying to make sure that by October, by January, there is further support and what the Government will be doing, whoever is the prime minister, is making sure there is extra cash to help people.
“I think it is crucial to understand that, first of all, we realised for a long time that things were going to be tough and that’s why we put in place a lot of measures already.
Work and Pensions Secretary Therese Coffey earlier also sought to downplay fears over energy prices, claiming they will be “nothing near” the forecasts.
Auxilione’s forecast released on Friday predicted the price cap on energy bills could reach £3,628 in October, from £1,971 today. They add it could then rise again to £4,538 in January and peak at £5,277 in April.
The worst forecast yet comes just 24 hours after Auxilione said regulator Ofgem could be forced to raise the price cap for the average household to £5,038 from next April.
Increases to the price of gas and electricity on wholesale markets are worsening the predicted rises for the cap, but they will fall if energy prices decrease.
It comes as gross domestic product (GDP) fell by 0.1% between April and June, according to new figures from the Office for National Statistics.
Mr Zahawi told Sky News: “My message to families today is: we will have those options ready to go.
“Yesterday I met with the industry to look at what more we can do with the industry on direct debit, on pre-payment meters, all the things that families are worried about, we’re making sure we’re doing the work so on September 5 the new prime minister can hit the ground running and get those things into place.”
Ms Coffey also told ITV’s Good Morning Britain: “Just like happened in May, we waited for Ofgem to come out with the formal changes on what might happen to energy prices or what would happen with the price cap.
“All the figures I have seen do not in any way suggest an average energy bill next year of £5,000, nothing near like that.”
Former chancellor Mr Sunak has said he is prepared to find up to £10 billion of extra support – with a vision to cover the total cost for up to 16 million vulnerable people, according to The Times.
Ms Truss has stood by her tax-cutting plans, saying it is her “first preference” before considering other measures.
Labour leader Sir Keir Starmer labelled Mr Johnson a “lame duck”, telling reporters during a visit to Scotland: “We do need a strategic, a credible, plan and that’s exactly what’s missing from this Government.
“For weeks now, and for weeks to come, we’ve seen the spectacle of two leadership hopefuls fighting each other, internal battle, actually arguing about just how bad they’ve been in Government and a Prime Minister who is a lame duck because he recognises there’s a problem with energy bills, but says I’m not going to do anything about it.
“So, yes, we need a strategic plan, and that’s what I will be setting out on Monday.”