how do people access finance in the emerging economies?

To take Nigeria as an example of how difficult it is to get access to credit for a small business:

All around the world, businesses struggle to access the financing they need to expand. ... While most respondents can access a bank, the banks themselves have a role to play in outreach and reducing the perception of high collateral requirements.

Access to Finance in Nigeria - GeoPoll

Here's a government report looking at 'financial inclusion':

Financial Inclusion In Nigeria:Issues And Challenges

It's difficult though, as with this report from last summer:

Nigeria Misses Goal on Growing Access to Financial Services

Almost 36% of adults don’t have a formal bank account
Government had sought to reduce that to just 20% by 2020

Nigeria fell short of its goal to bring more of its citizens into the regulated financial system, with the effects of the pandemic and difficulty in penetrating rural areas weighing on the effort.
Almost 36% of adults in Africa’s most populous country didn’t have any kind of bank account at the end of 2020, according to EFInA, a U.K.-backed development organization that seeks to bolster inclusive finance in Nigeria. The figure was little changed from two years ago and well above the government’s goal of cutting the proportion of nationals without financial access to 20%, which it set in 2013.
The West African nation has sought to bring more of its citizens into the formal financial sector as part of efforts to modernize its economy, bolster tax collection and cut back on informal jobs that often exploit workers. It has licensed banks, wireless carriers and technology companies to offer services as part of efforts to broaden access, especially for the two-thirds of the population that live outside of cities. 

Nigeria Misses Goal on Growing Access to Financial Services - Bloomberg

Perhaps we need to go to 'fintech':

Harnessing Nigeria’s fintech potential

How stakeholders could position the fintech sector for growth now and beyond the crisis.
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Banking in Nigeria remains an attractive sector, with over $9 billion in value pools, but despite high levels of competition, the vast majority of consumers are underserved. Lack of access to services, especially in rural areas, issues of affordability, and poor user experience all contribute to the frustration consumers experience right across the customer spectrum.
This has created an opening that fintechs have been quick to take advantage of, with many stepping up to develop enhanced propositions across the value chain to address pain points in affordable payments, quick loans, and flexible savings and investments, among others...

Fintech in Nigeria | McKinsey

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