The Central Park Obelisk has a long history as far as monuments go. The origins of the monument go back to the reign of Thutmose III in the New Kingdom. He erected the Central Park Obelisk and the London Obelisk (both called Cleopatra's Needles) in Heliopolis, modern-day Cairo. In the later part of the New Kingdom, king Ramesses II inscribed his own names upon the obelisk, but notably did not erase the name of Thutmose III.
The next notable stage of the obelisk's history is a thousand years later, during the reign of Augustus in 12 BCE. The two Needles were moved from their location in Heliopolis to the city of Alexandria to reside in the Caesarium. The two obelisks suffered a certain degree of degradation in this area due to the higher water table and the Mediterranean air, but they were still in rather good shape when acquisitioned by Britain and America.
When Europeans began filtering into Egypt in the 18th century, relatively little attention was given to the obelisks in comparison to their Greco-Roman counterparts. This etching was drawn by BenoƮt de Maillet in 1735. Rather than take the Egyptian obelisk, Mallet considered Pompey's pillar, depicted to the right, as the monument that France should focus upon acquiring.
A hundred years later, in a greater height of Egyptomania, European countries finally had a great desire to acquire these monuments to boost their own prestige. After one was taken by both Britain and France, the United States pushed for the American Consul in Cairo to acquire an obelisk as well for New York City. Khedive Ismail Pasha agreed to gift the monument to the US in the hopes of securing greater trade relationships. The transportation costs were covered by William Vanderbilt and the monument was erected in Central Park in 1881.
By this time, the monument had already suffered a degree of degradation on two of its sides. This engraving made around the time of its erection in NYC shows two of its sides suffering weathering damage, while the two others were still quite legible. During conservation studies taken on the obelisk in recent years, it has also been found that part of the tip was broken off and repaired by someone with a small metal pin. In an interview, Bob Brier, an Egyptologist who was involved in the monument's conservation suggested that this damage may have taken place during the transportation from Egypt to New York but never reported.
Since its erection in Central Park, the obelisk has suffered a great degree of degradation from pollution and New York weather. Rainwater at times seeped into cracks of the monument and froze, chipping off granite pieces from the obelisk. Acidic rain has also heavily contributed to weathering of the granite. Several decades ago, an effort was made to coat the granite in wax, which did a great deal in conserving the obelisk. Unfortunately, this was not done when it was first erected in Central Park as was done to the London obelisk.
I have yet to find evidence of Egyptian authorities requesting or suggesting the repatriation of the monument in the twentieth century. However, in 2010, Zahi Hawass sent a letter to the Central Park Conservancy to pressure them to take greater conservation efforts upon the degrading monument. The Conservancy seems to have followed Hawass' recommendation and undertook an extensive conservation project upon the monument.
Further Reading:
https://www.centralparknyc.org/locations/obelisk
https://web.archive.org/web/20141109191722/http://www.drhawass.com/blog/obelisk-central-park
https://octavianreport.com/article/bob-brier-cleopatras-needle/
Whose Pharaohs? by Donald Malcolm Reid