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The Sandgrounders are just six points behind the Blues in the Vanarama Conference, and Burr's men have won only once in their last nine games in all competitions.
The Chester boss has shaken up his squad since the January transfer window opened, signing striker Cory Williams, goalkeeper Jon Viscosi and right-back Ryan Higgins, and completing loan deals for Fleetwood Town midfielder Josh Green, Aston Villa striker Thomas Strain and Barnsley midfielder Brad Abbott.
He remains hopeful of signing and another striker and wants to see the Blues begin their climb from 17th in the table when they head to the Merseyrail Community Stadium this weekend.
Burr said: "We want to try and push ourselves into that top half. We had a disappointing Christmas, albeit the Halifax result was a tremendous result away from home.
"We've built ourselves on good home performances and let ourselves down with the Braintree and Altrincham games but sometimes that can happen.
"I want us to push on into the top half, that's what I'm looking at and we're capable of doing that.
"We've shown already this season that we're capable but it's important that as a group we work hard and get back to the good things that we did to get us the points that we've got already."
Southport boss Gary Brabin and his assistant Paul Carden need no introduction to Blues fans as the pair played for Chester City and both won the club’s player of the season award.
Brabin, in his second spell in charge having replaced Martin Foyle in October, has challenged his team to build on an excellent run to the FA Cup third round, which ended with a late 1-0 defeat at Championship outfit Derby County. He has brought Andy Parry back to the club on a month's loan from Luton Town and extended the loan of Walsall striker Amadou Bakayoko.
Brabin said: "We don’t go into any games thinking about a draw. We’d like to win on Saturday and I think we can do.
“We’ve tried to get rid of any negativity and focus on ourselves, rather than our opponents. I want the team to enjoy their football and I think our recent results reflect that.”