Christine Chapman, of the assembly's Communities Committee, said good work was being done - but the Welsh government should speed up efforts.
The latest figures suggest more than one in five people in Wales are living in poverty.
The Welsh government said getting people out of poverty was at the "very heart" of what it does.
Earlier this year, the Communities Committee published a report that stated it was "deeply concerned by the Welsh government's lack of progress in reducing poverty".
Poverty is statistically defined as a household income below Â£271 a week - 60% of the national mid-point average of Â£453.
Figures from the Joseph Rowntree Foundation suggest 23% of people in Wales are living in poverty - a figure which the committee said has barely changed since the early 2000s.
Labour AM Christine Chapman said the Welsh government was "going in the right direction" but the pace of change "needs to be quicker".
"We are a rich country in the main but yet people are still struggling with their heating bills, with their food - that is absolutely appalling," she said.
"There must be a strong message first of all to the UK government that things need to change."
Gemma Jones, a caseworker for Rhondda Cynon Taff Citizens Advice, said despite government initiatives, she sees little evidence of improvement.
"I don't know what's going wrong or what's happening but we're certainly seeing a lot more clients in poverty and not saying I'm better off," she said.
A Welsh government spokesman responded: "All government departments are working relentlessly to tackle the root causes of poverty, by giving children the best possible start in life, helping people to get a job and improving their health and wellbeing.
"We also continue to do all we can to mitigate the effects of the UK government's sweeping welfare reforms and austerity measures.
Sunday Politics Wales is on BBC One Wales at 11:00 GMT on 13 December

Arteta, 34, joined the club in 2011 and made 150 appearances, scoring 17 goals.
Rosicky, 35, signed in 2006 and was given an award for 246 appearances and 10 years of service before the Gunners' final-day win over Aston Villa.
Manager Arsene Wenger described the pair as "two exceptional players and influences in the dressing room".
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Spaniard Arteta and Rosicky of the Czech Republic leave Arsenal with two FA Cup medals, won in 2014 and 2015.
Arteta, who signed from Everton for £10m, thought he had scored on his farewell but his late strike in the 4-0 win over Villa on Sunday was given as a Mark Bunn own goal.
He was then seen crying after the final whistle.
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Tottenham's 5-1 defeat by Newcastle meant the Gunners leapfrogged their north London rivals to finish second in the table, but they have not won the title since 2004.

Allegri, 48, took charge in July 2014 and signed a new deal until 2017 after leading the Old Lady to the league title and the Coppa Italia.
Having successfully defended their league crown, Juventus have announced the former AC Milan boss will now remain until 30 June 2018.
Juventus face AC Milan in the Coppa Italia final on 21 May.
They are 12 points clear of nearest rivals Napoli with two games remaining in Serie A.
Juventus defeated Manchester City twice in the group stage of this season's Champions League before being knocked out by Bayern Munich in the round of 16.
More urgency is needed to tackle poverty in Wales, a Labour assembly member has said.
[NEXT_CONCEPT]
Arsenal captain Mikel Arteta and fellow midfielder Tomas Rosicky will leave the Premier League club when their contracts expire this summer.
[NEXT_CONCEPT]
Juventus boss Massimiliano Allegri has extended his contract with the Serie A champions until the summer of 2018.