Probably not, but if there is it will need a better study that the one by MigrationWatch to demonstrate it.

The trouble with proving anything in Economics (which is what this is really about) is that the theory is imperfect and there are many alternative views and theories with too few good experiments to differentiate them.

But the popular view about why immigration will cause unemployment is based on what most economists agree is a fallacy called the "lump of labour" fallacy (see the definition on the website of The Economist and this NYT article by Paul Krugman). The basic idea is that there is a fixed pool of work to be done. Therefore, if there are more people, there will be higher unemployment as there won't be enough work to go around. The idea probably goes back to the Luddites who smashed the machines that made factories more efficient apparently threatening their jobs.

Most economists think this is nonsense. Not least because, observationally, it hasn't happened that way. Since the industrial revolution the economies of most countries have become immensely more productive as we substitute machines for (less productive) people. But most people are not unemployed despite being "replaced" by machines and most countries are far more wealthy then they were in 1850. Higher productivity makes societies wealthier; wealthier societies can afford more work so there are more jobs.

So the public's intuition is based on a simplistic theory.

But theory is rarely enough to settle an argument in economics where the complications of the real world often lead to observations that are hard to explain in theory. So what do the empirical studies suggest about the UK?

First the one that caused the controversy. This is by MigrationWatch, a broadly anti-immigration pressure group with the ear of some members of the current government (and this is the study referred to in the question).

But the lump of labour fallacy is very intuitive but few experts would defend it.

In short their approach to the issue involved rather more statistical analysis than MigrationWatch and they made a serious attempt to separate out confounding factors. They achieved this by looking at the unemployment and migration statistics in local authority regions. They also used National Insurance Number (NINo) registrations to measure migrant arrivals which is thought to be a much more reliable metric than the survey-based data used to estimate the gross numbers. This allowed them to compare local changes in immigration to local changes in unemployment giving a lot more insight than a gross country-wide number.

The results show a very small negative and generally insignificant correlation between the migrant inflow rate and the change in the claimant count rate...  ...Perhaps surprisingly  , the interaction between migrant inflows and GDP emerges as positive and significant, albeit very small, indicating that during periods of lower growth, migrant inflows are associated with a lower claimant rates relative to the counterfactual - in other words, slower claimant growth than would otherwise have occurred.

Summary

In short, there is probably no link between youth unemployment and immigration but, if there is, MigrationWatch will need to produce a better study to demonstrate the existence of such a link.

The trouble with proving anything in economics, which is what this is really about, is that the theory is imperfect and there are many alternative views and theories with too few good experiments to differentiate them.

The popular view about why immigration will cause unemployment is based on what most economists agree is a fallacy called the "lump of labour" fallacy. (See the definition on the website of The Economist and this NYT article by Paul Krugman). The basic idea is that there is a fixed pool of work to be done. Therefore, if there are more people, there will be higher unemployment as there won't be enough work to go around. The idea probably goes back to the Luddites who smashed the machines that made factories more efficient apparently threatening their jobs.

Most economists think this is nonsense. Observationally, it hasn't happened that way. Since the industrial revolution, the economies of most countries have become immensely more productive as we have replaced people by more productive machines. Yet, most people are not unemployed and most countries are far wealthier than they were in 1850. Higher productivity makes societies wealthier; wealthier societies can afford more goods so there are more jobs.

Basically, modern mainstream economics does not agree with the public's intuition.

However, theory is rarely enough to settle an argument in economics, where the complications of the real world often lead to observations that are hard to explain in theory. So what do the empirical studies suggest about the UK?

The ones that produced the study, MigrationWatch, is a broadly anti-immigration pressure group with the ear of some members of the current government (and this is the study referred to in the question).

The lump of labour fallacy is very intuitive but few experts would defend it.

In short, their approach to the issue involved rather more statistical analysis than MigrationWatch and they made a serious attempt to separate out confounding factors. They achieved this by looking at the unemployment and migration statistics in local authority regions. They also used National Insurance Number (NINo) registrations to measure migrant arrivals which is thought to be a much more reliable metric than the survey-based data used to estimate the gross numbers. This allowed them to compare local changes in immigration to local changes in unemployment giving a lot more insight than a gross country-wide number.

The results show a very small negative and generally insignificant correlation between the migrant inflow rate and the change in the claimant count rate.  ... Perhaps surprisingly, the interaction between migrant inflows and GDP emerges as positive and significant, albeit very small, indicating that during periods of lower growth, migrant inflows are associated with a lower claimant rates relative to the counterfactual - in other words, slower claimant growth than would otherwise have occurred.

Summary

Migration Watch also claim that findings of no significant effect "defy intuition". But the lump of labour fallacy is very intuitive but few experts would defend it.

Migration Watch also claim that findings of no significant effect "defy intuition". In their exact words (paragraph 5, my emphasis):

These results have been criticised on the grounds that they are counter-intuitive, given the scale of A8 migration into the UK and the characteristics of the migrants.

But the lump of labour fallacy is very intuitive but few experts would defend it.