The vines are easy to redirect. Lift up the ends, bend them modestly in the direction you want them to go and place them down. If you need to you can push a stake in the ground to prevent vines from snapping back into position. Once placed like this they tend to continue in the new direction. But this only goes so far, you're in a losing battle. Forget about mowing under or near them. Just concede that the summer and fall belong to them. Just accept a no-mow zone as the price of pumpkin stewardship. Or you can always grow carrots instead.

I always mow under strong climbers: let vines grow two yards into the lawn and cut off the nodal roots before mowing. Act cautiously, when you lay the vines all in one direction alongside the patches, where they come from. And after the mowing you lay patterns as you like. Here my son shows you a V-pattern, where vines from two sides were newly laid a month before and now because of the fruits there was not mowed again.