Thousands have taken to the streets across the Caribbean nation since Thursday calling for Mr Moise to step down.
Clashes between protesters and police took place over the weekend, with local reports claiming as many as four people have been killed.
At least one person was fatally shot, apparently by nearby police officers, during a demonstration in downtown Port-au-Prince on Saturday.
In the city’s Petionville neighbourhood protesters lit fires and blocked the road to Mr Moise’s house, throwing stones at the property after guards protecting a presidential ally had hit a woman’s car before beating her.
Injuries were also reported during a clash in Mirebalais, around 25 miles outside the capital, on Thursday.
The Haitian National Police (PNH) said a number of its officers had been hurt at the demonstration, while announcing it was aware at least two people died during the protests.
“Several agents were victims of malicious individuals who had infiltrated the event, wanted to create panic and chaos by throwing stones and shooting for no apparent reason,” a PNH statement read.
“Police and police officers were able to exercise restraint and level-headedness in Mirabalais, where the PNH had nine wounded in its ranks without a victim.
“The high command is proud of their behaviour and encourages them to control their emotions so as not to succumb to provocation.”
Protesters have called for Mr Moise’s resignation over allegations of corruption and financial mismanagement, while the president has offered to engage in talks with opposition leaders.
Demonstrators are angry about skyrocketing inflation and the government’s failure to bring prosecutions over alleged embezzlement during a Venezuelan programme that sent discounted oil to Haiti.
The United Nations Mission for the Support of Justice in Haiti (MINUJUSTH), a peacekeeping mission in the country, called for democracy to take place “through the ballot box, not through violence”.
Representative nations from the group urged the Haitian government to pass electoral and budget legislation that would pave the way for “free, fair and transparent” elections to take place as scheduled in October.
“[MINUJUSTH] deplores the loss of life and property damage caused by the unacceptable acts of violence that took place on the margins of the rallies, while acknowledging the professionalism demonstrated by the Haitian National Police as a whole,” a statement from the group read.
“The group members call on the Haitian society actors, and primarily the country's leaders, to engage in a constructive and inclusive dialogue in order to identify and implement realistic and lasting solutions to the political and economic crisis currently occurring in Haiti.”
Additional reporting by AP