It's not going to be fuel starvation due to a tank breather.
Most carbs have a slow run system and a main system. With a small engine, it's likely that the main system comes into action fairly low down in the rev range. If only tick-over and slightly beyond is dodgy, then it's a fair bet the slow run system is at fault. I'm not quite sure about you description. Is the tickover lumpy and the progression fine or it the tickover fine and the progression lumpy?
Either way, a couple of quick fixes for you. Open the mixture screw fully or even remove it completely and put your finger against the hole to seal it. Start the engine, if you can get it to go. Even a reasonable blurp might do it. What you've actually done here is open up the slow run system on the down side of the jet and any small blockage will likely be sucked through. However, if the tickover is fine and the progression lumpy, this is not likely to work. Alternatively, you can try this since you are able to rev it; take off the air box, rev the engine then snap the throttle shut and immediately cover the intake to the carb with your hand. This will put the full depression of the engine onto the jets and it is a good way of removing a partial blockage.
However, if the carb has an idle jet and it is blocked by a relatively large particle on the float chamber side, the above may not be able to suck the debris through the jet. In this case, it'll be a carb off clean out job.
Since I'm not familiar with your particular engine, there is a caveat. Some tickover mixture adjustment screws adjust the air rather than the fuel. If this is the case, the above it less likely to work...but it might.
For completeness, if the the tickover is fine and fairly high speed is also fine, then it is likely a blockage at a progression hole. A progression hole is a small drilling in the carb which feeds some fuel as the throttle is opened. There may be more than one of them. As the throttle opening progresses, the main system comes into play and the tickover/progression system ceases to function. The hand over the carb technique will attempt to clear a progression problem.
I hope you can follow all this. It's a bit late for complex explanations plus, the whiskey doesn't help.