The charity's comments come after the government released unemployment, wage and inflation figures for the last quarter.
Unemployment had fallen to 6% - its lowest level in six years.
Wage increases were also higher than prices rises. If bonuses are included, wage increases were 1.6% higher than price rises. Without bonuses, wage increases were 1.4% higher.
It's the first time wage increases both including and excluding bonuses were higher than inflation for six years.
Paul Archer, from CAP, told Premier: "Growth is usually going to happen for those who are already on a fairly good income.
"We need to start seeing those who are in the most severe poverty being helped.
"The kind of challenge now is making sure there are a lot of people across the UK that are helping the worst off, because they're the ones that probably won't have seen the benefit... of the economic upturn.
"The thing people need is just time... someone [who] is going to invest them, who is going to care for them, who's going to believe in them.
"If you've had rejection letters pile up it's very hard to believe in yourself. But if you've got a friend that can encourage you and support you, that's going to make a massive difference.
"We are facing an amazing opportunity to help the poor through local churches... there's a great opportunity there for Christians and local churches much more holistically."
Listen to Paul Archer speaking to Premier's Aaron James:
