The singer also dedicated a song to her husband, Blake Fielder-Civil, who is being held on remand pending charges including GBH.
During the show, the 24-year-old told the crowd: "To them people booing, wait 'til my husband gets out of incarceration. And I mean that."
Some fans have contacted the venue and tour promoter to ask for refunds.
Winehouse's spokesman said the singer had had a particularly hard day on Wednesday after visiting her husband in prison for the first time.
He added it was completely Winehouse's decision to go on stage as she knew so many people had paid to see her.
The tour is due to continue as planned on Friday night in Glasgow.