Avenue conceded in the last minute with ten men to lose their opening home game, before that they had won their first away game due to a 97th minute Adam Boyes strike, writes Tom Feaheny.
Bower says that the performances and work rate have impressed him and they can use that going forward.
"We have shown we can create chances, we have withstood pressure, we will work hard and get them ready for the next game.
He added: "We have players who have played together at other clubs, we have shown good understanding and link-up play, we are still work in progress, we just want to keep on improving."
Their next game is the all-ticket clash between themselves and York City in the Vanarama National League North tie.
Bower believes that the Minstermen will be challenging to get back to the Vanarama National League via automatic promotion.
"They are one of the two I expect to be challenging it will be a big test for us, I expect a good crowd, if we can play like we did against Southport in patches, we can win it."
Picture by Callum Ronan Creative