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but I have not found a way to add a condition.

You can add the condition in you model function firstCheckedChecklistItem() or inside with function itself

public function firstCheckedChecklistItem() { 
    return $this->hasOne('App\Models\CheckedChecklistItem')
         ->whereNotNull('done_by')
         ->latestOfMany(); 
}

  ->with(['machineType.checklistGroup.currentChecklists.checklistItem.firstCheckedChecklistItem' => function($query) {
       $query->whereNotNull('done_by');
  }])

Reference: How to access model hasMany Relation with where condition?

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