Hi Robertskitty,
I actually think that your mesh and tank hood are a safety measure against axies jumping out and also aerosol or particulate pollutants entering your tank and fouling the water. Personally i would not remove them.
I guess there would always be a certain degree of inconvenience in siphoning wastes. I can only think of some ways that can help make it less bothersome.
1 - Feed less messy food types like live earthworms or blackworms over frozen bloodworms or pellets.
2 - Try hand-feeding with long feeding forceps so you can make sure your axie eats the food and not left it to rot in the water.
3 - Don't overfeed.
4 - You may like to change your tank to bare bottomed if you are currently using sand.
5 - Use simple easy access hiding places like pipes and pots over ornate aquarium decorations or rocks that may cause food and wastes to accumulate between the spaces.
6 - Use an aquarium siphon over a turkey baster. I find that siphons are easier to use and more effective. However, it really depends on what you are more comfortable using.
Sorry its not much of a direct help.
Cheers