It seems obvious to me that fewer low skill workers would result in higher wages for those who could most benefit from it.
Here's one classic study on the effect:
David Card, "The Impact of the Mariel Boatlift on the Miami Labor Market" (1990), http://davidcard.berkeley.edu/papers/mariel-impact.pdf
Quoting from the abstract: "…This paper describes the effect of the Mariel Boatlift of 1980 on the Miami labor market. The Mariel immigrants increased the Miami labor force by 7%, and the percentage increase in labor supply to less-skilled occupations and industries was even greater because most of the immigrants were relatively unskilled. Nevertheless, the Mariel influx appears to have had virtually no effect on the wages or unemployment rates of less-skilled workers…"
That's a rapid influx of 7% of Miami's population! But the effect isn't obvious to economists, either, and you can find people arguing both sides. This is a fairly balanced article: https://www.npr.org/2017/08/04/541321716/fact-check-have-low...
For example: https://www.migrationpolicy.org/sites/default/files/source_i...
So yes, it is likely that immigration rates have a negative effect on the wages of native workers in low barrier of entry positions. You'd have to suspend disbelief to accept the narrative that there is no impact.
source: https://www.migrationpolicy.org/article/frequently-requested...