Franklin D. Roosevelt certainly holds a special place in the history of the United States. Roosevelt is the only President to serve more than two terms (he served four), and he guided the United States through some very trying times, including the heart of the Great Depression and the outbreak of World War II. FDR's second term certainly saw him face a number of challenges, some of which had favorable outcomes and some of which did not.
One of the major failures of FDR's second term would be the defeat of Roosevelt's "court packing" plan. To understand this we have to look at FDR's first term. During his first term, he attempted to address the problems of the Great Depression by creating a series of programs called "The New Deal." The New Deal aimed to put Americans back to work through the use of public works projects, as well as ensure that elderly and sick Americans were taken care of. Under the New Deal, workers would also be hired by government agencies to do jobs such as the construction of pools, roads, schools, dams. Roosevelt's programs, however, faced opposition from the US Supreme Court on the grounds they were unconstitutional and should not be enacted by the federal government.
In order to deal with the opposition of the Supreme Court, Roosevelt introduced a bill which would allow for an expansion of the Supreme Court. The bill proposed adding one new justice for each justice over 70 years old with potential for up to six new justices. Roosevelt hoped that this legislation would allow him to appoint new justices that would ultimately shift the court to a more supportive stance regarding his New Deal programs.
After debating the issue the U.S. Congress decided to reject Roosevelt's "court packing" bill. Roosevelt would not be able to appoint additional justices to reshape the Supreme Court. The failure of Roosevelt's "court packing" bill can certainly be seen as a failure of his second term.
The Supreme Court did, however, uphold Roosevelt's controversial Social Security Act, which was seen as a rare shift in opinion by the court. This saved Roosevelt's Social Security program and can be seen as an achievement during his second term.
Another failure of Roosevelt's second term can be seen in his attempts to shift the political ideology of the Democratic Party. In 1938 Roosevelt encouraged the challenging of more conservative members of the Democratic party that were not fully behind his New Deal programs. Roosevelt sought to replace them with more supportive liberal members of the Democratic Party in order to ensure greater support for his policies. He hoped to do this through the Democratic Party primary elections. This proved to be a failure for Roosevelt as many of the candidates he backed were defeated. The 1938 general elections led to an increase in the number of Republican and conservative Democrats in Congress. With the Republicans and conservative Democrats forming an alliance, Roosevelt faced stronger opposition in Congress.
I would argue that another accomplishment to be found in Roosevelt's second term would be his handling of international politics regarding the beginning of World WarII. Near the end of Roosevelt's second term, he began to abandon old isolationist policies and began to provide assistance to the allied powers, particularly Britain and France (until their defeat). Roosevelt would eventually further expand on this support for Britain while also beginning to build up the military power of the United States in preparation for potential entry into the conflict. Roosevelt's efforts helped to ensure that the dangerous governments of Japan, Germany, and Italy would not achieve the domination they desired and also helped to secure the safety of the United States.