"Just out of curiosity why is there a need for a path between that particular circle and the cement path?"
It's time to preserve our park's history. We should no longer settle for what we are given after 25 years of neglect by the city. It was poor planning that allowed the President's House at Richard Bland College to gain an artful 19th century fountain while we are forced to accept a mid-20th century one in a different place.
1903 picture (looking southeast toward Fillmore & South Adams)
Current Fountain (original fountain location in background)
However, back to the question. In the short term, this path is the last of the sections of original paths between Sycamore Street and the 1967 concrete addition that is without gravel.
In the long term it is part of the interlocking path systems that must be fully restored with gravel. Furthermore, in the 1980 amended nomination for the National Historic District, it specifically list the paths of the park as a contributing feature. Quoted text follows:
"The last and most significant improvements of Poplar Lawn occurred in 1875. At this time funds were appropriated for the planting of five hundred trees on Poplar Lawn at the cost of one hundred dollars. It was probably during this year also that irregular paths were laid out in rough concentric and radiating patterns. The plan of the park reflects the popularity of landscape theories practiced by Frederick Law Olmsted during the second half of the 19th century."
True Poplar Lawn is not an Olmsted, but this revolutionary planning is important in recognizing the history of how Petersburg saw itself.
The largest city in the US in 1870 was New York City. Petersburg is listed as the 77th largest city in the US, but having a park referred to as "Central Park" and laid out in an Olmsted-inspired pattern probably made Petersburg feel they were number 1.
There is no reason Petersburg should not be able to recapture that spirit to feel number 1 again.