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You can skip some steps from Erik's answer if you use the tool "Overlap analysis". This works if all input layers are vector layers. Choose your grid as input and the cover layer as overlay. The result is a grid layer (spatially identical to your input) with the additional information on coverage (area & percentage) of your input by your overlay. Then you can filter, delete, ... features depending on your cut off value (check step 6 of Erik's answer).

EDIT: below screenshots of an intial grid and a filtered grid (coverage area > 50%)

You can skip some steps from Erik's answer (step 2 to 5) if you use the tool "Overlap analysis". This works if all input layers are vector layers. Choose your grid as input and the cover layer as overlay. The result is a grid layer (spatially identical to your input) with the additional information on coverage (area & percentage) of your input by your overlay. Then you can filter, delete, ... features depending on your cut off value (check step 6 of Erik's answer).

EDIT: below screenshots of an intial grid and a filtered grid (coverage area > 50%)

You can skip some steps from Erik's answer if you use the tool "Overlap analysis". This works if all input layers are vector layers. Choose your grid as input and the cover layer as overlay. The result is a grid layer (spatially identical to your input) with the additional information of coverage and coverage percentage of your input by your overlay. Then you can filter, delete, ... features depending on your cut off value (check step 6 of Erik's answer).

EDIT: below screenshots of an intial grid and a filtered grid (coverage area > 50%)

You can skip some steps from Erik's answer if you use the tool "Overlap analysis". This works if all input layers are vector layers. Choose your grid as input and the cover layer as overlay. The result is a grid layer (spatially identical to your input) with the additional information on coverage (area & percentage) of your input by your overlay. Then you can filter, delete, ... features depending on your cut off value (check step 6 of Erik's answer).

EDIT: below screenshots of an intial grid and a filtered grid (coverage area > 50%)

You can skip some steps from Erik's answer if you use the tool "Overlap analysis". This works if all input layers are vector layers. Choose your grid as input and the cover layer as overlay. The result is a grid layer (spatially identical to your input) with the additional information of coverage and coverage percentage of your input by your overlay. Then you can filter, delete, ... features depending on your cut off value (check step 6 of Erik's answer).

You can skip some steps from Erik's answer if you use the tool "Overlap analysis". This works if all input layers are vector layers. Choose your grid as input and the cover layer as overlay. The result is a grid layer (spatially identical to your input) with the additional information of coverage and coverage percentage of your input by your overlay. Then you can filter, delete, ... features depending on your cut off value (check step 6 of Erik's answer).

EDIT: below screenshots of an intial grid and a filtered grid (coverage area > 50%)