I know they like Chufa. Around here it is mostly Soy beans or corn, fresh planted they dig it up. Cruise green grass for bugs. Hens also nest in thick grass so I wouldn't mow down all your grass.
If they are passing through your land from where they roost to the neighbors, will be no big issue killing one in transit. In the Spring toms range around looking for hens esp a couple hours after first light or right away if not roosted with hens. I have seen them go a mile or more just cruising around.
If you can set up fairly close to the roost in between them and where they want to go, a little calling with a hen/ jake pair decoy should have you with one in range eventually.
