Ministers are working to ensure that "everything we can do to maximise the continuity of these small businesses is done", he said.
Chaired by Clark, with support from the small business minister Andrew Griffiths, the initiative's attendees included representatives from business bodies, the construction trade sector, unions, banks and government.
But Mr Lidington was also forced to defend the Government's actions amid anger at ministers' relationships with Carillion.
Lloyds said it wanted to provide "reassurances" at an "unsettling" time.
Mr Corbyn told the Guardian he wanted to stop "middlemen creaming off the profits" by blocking the Government and local authorities from outsourcing services unless they were failing.
It will act as a means to work together to ensure the impact of the Carillion insolvency on the company's employees in the private, as well as public, sector is minimised and to help them recover.
Smaller businesses that were subcontracted by Carillion are expected to bear the brunt of the company's collapse.
Lloyds said small business customers in Carillion's supply chain will be able to apply to have fees on overdrafts and invoice finance facilities waived or for capital repayment holidays on existing loans.
"There should be no equivocation from the Government and no attempt to duck and dive around this issue".
Speaking about the launch of the fund, Amanda Murphy, Head of Commercial Banking for HSBC UK, said: "We have set up an emergency fund to help any Carillion supplier facing financial difficulty and will make decisions taking recent events into full consideration".
TUC general secretary Frances O'Grady welcomed the formation of the task force.
These included work on the HS2 railway, the construction of new schools and hospitals, the building of the Aberdeen bypass and catering contracts with both schools and the Ministry of Defence.
The company had about $1.7 billion in debt and an $800 million pension deficit.
"All of the Carillion rail works on the various contracts can be brought in-house easily and that is what we expect to happen with jobs and services protected".
Similarly, ex-finance chief Zafar Khan, who left the company in September, will receive £425,000 in salary for 12 months; while interim chief executive Keith Cochrane will be paid his £750,000 salary until July, despite leaving Carillion in February.