Politifact seems to have done a decent rundown on this one explaining the details.
It's technically true-ish (2009, not 2011 etc) but it's hard to call it a "cost", they've been getting partial refunds on taxes they've paid for children who are mostly American citizens.
The $4.2 billion figure caught our eye. Though Trump’s campaign didn’t
get back to us, we found a 2011 audit by the Treasury Inspector
General for Tax Administration that corroborates his claim.
Nonetheless, Trump confuses a few points.
In 2009 (not, as Trump says, in 2011), "individuals who are not
authorized to work in the United States were paid $4.2 billion in
refundable credits, according to the report. That’s indeed more than
four times the amount in 2005 ($924 million).
How did this happen?
Although undocumented immigrants can’t get a Social Security Number,
they can file taxes with a different nine-digit number, the Individual
Taxpayer Identification Number or ITIN.
Experts told us that illegal immigrants make up the vast majority of
ITIN filers, though there’s no way of knowing just how many. The group
also includes legal immigrants (refugees, asylum seekers, spouses),
foreign workers (professors, technology workers, people who own
businesses in America but live abroad), and dependents of both
citizens and immigrants.
Regardless of immigration status, ITIN filers are not eligible for
Social Security. According to the Social Security Administration,
undocumented immigrants doled out an estimated $12 billion in payroll
taxes but will never get the benefits. Also, ITIN filers can’t get the
Earned Income Tax Credit, due to a provision in a 1996 law.
ITIN filers, however, can receive the Additional Child Tax Credit, a
refund given to people who owed less in taxes than deductions they
could receive through the Child Tax Credit (up to $1,000 per child).
The credit as it stands today was established in the Economic Growth
and Tax Relief Reconciliation Act of 2001, one of the tax cuts passed
under President George W. Bush. Unlike in 1996, Congress did not write
a provision barring ITIN filers from claiming the refund.
As a result, claims for the additional child credit have increased
significantly since 2001, according to the Treasury Inspector General
audit. By 2009, 2.3 million ITIN filers received $4.2 billion through
the additional child credit, a four-fold increase over 2005. Here’s a
chart from the audit:
Under this current system for ITIN filers, "the government isn’t
losing out," according to Bob Greenstein, president of the
left-leaning Center on Budget and Policy Priorities. It gets more
money in revenues than it gives out in credits.
However, Greenstein told us it’s slightly misleading to say that the
recipients of the child credit are illegal immigrants.
"The vast majority of that $4.2 billion, the filer may be
undocumented, but you have to have a child to receive it. And the
children are overwhelmingly U.S. citizens," he said.
In other words, the $4.2 billion in tax credits largely benefits
American-born children, whose parents are admittedly undocumented