Great question!

For those of you not familiar with this particular part of American political history, Huey Long was the famous populist governor of Louisiana, who turned his state’s constitution into silly putty, the entire state government into a giant patronage operation, and the only reason he was hated because of it was that, while he was absolutely corrupt on a personal level, he didn’t actually sell out to the powerful oil industry that had been running the state for decades.

Now, when Huey decided he’d like to become President on a third party ticket – after being U.S Senator and Governor essentially simultaneously – the centerpiece of his platform was the “Share the Wealth” Program, which included:

• various caps on wealth and income: personal wealth no higher than $5-8 million (or rather, no more than 300 times the average family wealth), annual income no higher than $1 million (or rather, more than 300 times the average family income), inheritances no higher than $5.1 million.
• sharply expanded progressive taxation on wealth and income.
• a guaranteed family income of $2,000-$2,500 (or rather not less than 1/3 of average family income). 
• a universal old age pension for everyone over the age of 60.
• government purchase and storage of surplus agricultural production.
• veterans pensions and health care.
• Free higher education and technical/vocational education.
• Public works to boost the economy.

Now you’ll note that a lot of this is actually quite similar to stuff Roosevelt was already doing or in the process of doing – public works, government subsidies to farmers, higher taxes on the rich, old age pensions, etc. – but just more so, mostly because Huey Long was primarily interested in outflanking FDR from the left. And other parts of it was just stuff that Huey Long thought would be popular.

And it’s that last part that colors how I feel about this: while I’m sure that he was quite genuine about wanting to help the poor and hurt the rich, Huey Long didn’t care whether he could pay for any of this, or whether he’d be able to get it through Congress, or whether the Supreme Court would consider any of it unconstitutional. because what he really wanted to do was defeat FDR in 1936 and then become President in 1940. 

And no, I don’t think it particularly pushed FDR to the left, because a lot of the things that people point to (Social Security, for example) were already in the works when Huey Long started pushing his program, and Huey was clearly taking parts of FDR’s program and saying “BUT BIGGER!” The biggest influence I would say it had on FDR was that he labeled his 1935 tax bill the “Soak the Rich Tax.”