Yes it is. However, whilst the Definitive Map Modification Order process is ongoing, the main Forestry Commission entrance to Hagg Wood from Intake lane will unfortunately remain closed. The current blockage has been in place since October last year when two new landowners took the action to the annoyance of the local community.
It is still possible for you to gain access to Hagg Wood from the Route 66 Cycle Path and then turn right if you are coming from Dunnington at the open field and follow the edge of the wood until you reach the public footpath sign (see Arrow 1) The sign is currently dislodged and leaning against a tree and has been reported to City of York Council. The other access point is via the Hagg Lane entrance (see Arrow 2).
However, you should still make sure that you observe the current Government regulations and guidance on social distancing and access to the countryside during the current coronavirus restrictions.
You can view a map of Hagg Wood at http://fohw.org.uk/publication2.htm