A sinkhole near the house is usually cause by water, like a nearby downspout dumping water too close to the building. Sometimes the answer is as easy as using a longer downspout, or may require and underground downspout.

Even though it looks as though the water would flow away from the building a level indicated that the ground was back pitched towards the house. The bushes and stepping-stones can create an optical illusion that the ground slopped away.
Accumulation of water next to a building will flow down to the weeping tiles and drain away to the city sewer. Every 100 gallons will take some soil with it and of course create a sinkhole.

This problem cannot be solved with a longer downspout because of the stepping stone pathway it would have to cross. This is the perfect situation for an underground downspout.

To install an underground downspout, remove the elbow at the base of the downspout. Dig a trench about 8-inches deep 10-feet out into the yard. The trench should pitch away from the house at about 1/8-inch per foot. Attach a debris filter to the building just below the downspout (to keep the pipe from filling with leaves). From there, use four-inch thin-wall PVC pipe to a tee at the end of the trench. Dig a posthole below the tee about 3-feet deep and fill it with pea-stone. From the top of the tee to ground level use PVC and a drain cover that can be removed for cleaning.

If there is a short rainfall the pea stone will absorb the water into the ground. If there is a heavy downpour the water will flow out of the drain cover out onto the lawn.