City were dominant in the first half as Sergio Aguero opened the scoring, but the game was more evenly contested after the interval and the Hammers carved out chances to draw level.
However, Yaya Toure's fabulous shot left West Ham with too much to do and they could only pull one goal back in stoppage-time thanks to Andy Carroll.
Asked if he felt West Ham could have got something from the match, Allardyce told Sky Sports: "At least we showed that we possibly could in the second half, because I was just a little bit concerned about the players showing too much respect to City in the first half and giving the ball straight back to them.
"We had to improve our play in possession and we persisted to do that throughout the whole game but, as our problem has been all season, for the chances created there are not enough goals scored.
"Our second-half performance was pleasing because they are a top-quality team who have spent millions of pounds and we managed to get to their level at times in the second half and cause them a few problems."
West Ham remain comfortably placed in mid-table despite Saturday's defeat but Allardyce is aware that he will need to bring in reinforcements in the transfer window to keep the club moving forward.
Questioned on what he would need to add over the summer, he replied: "Better players, it's as simple as that. Finding them and getting them is the hard bit, but if we can then we will get better and hopefully become an established Premier League side."