diff --git a/openmc/mesh.py b/openmc/mesh.py
index 560d54042..f3cd69a8f 100644
--- a/openmc/mesh.py
+++ b/openmc/mesh.py
@@ -2316,7 +2316,7 @@ def _read_meshes(elem):
         A dictionary with mesh IDs as keys and openmc.MeshBase
         instanaces as values
     """
-    out = dict()
+    out = {}
     for mesh_elem in elem.findall('mesh'):
         mesh = MeshBase.from_xml_element(mesh_elem)
         out[mesh.id] = mesh
diff --git a/openmc/settings.py b/openmc/settings.py
index b4e615a2a..6486d1f2a 100644
--- a/openmc/settings.py
+++ b/openmc/settings.py
@@ -1603,15 +1603,14 @@ class Settings:
                 if value is not None:
                     self.cutoff[key] = float(value)
 
-    def _entropy_mesh_from_xml_element(self, root, meshes=None):
+    def _entropy_mesh_from_xml_element(self, root, meshes):
         text = get_text(root, 'entropy_mesh')
-        if text is not None:
-            path = f"./mesh[@id='{int(text)}']"
-            elem = root.find(path)
-            if elem is not None:
-                self.entropy_mesh = RegularMesh.from_xml_element(elem)
-        if meshes is not None and self.entropy_mesh is not None:
-            meshes[self.entropy_mesh.id] = self.entropy_mesh
+        if text is None:
+            return
+        mesh_id = int(text)
+        if mesh_id not in meshes:
+            raise ValueError(f'Could not locate mesh with ID "{mesh_id}"')
+        self.entropy_mesh = meshes[mesh_id]
 
     def _trigger_from_xml_element(self, root):
         elem = root.find('trigger')
@@ -1671,15 +1670,14 @@ class Settings:
             values = [int(x) for x in text.split()]
             self.track = list(zip(values[::3], values[1::3], values[2::3]))
 
-    def _ufs_mesh_from_xml_element(self, root, meshes=None):
+    def _ufs_mesh_from_xml_element(self, root, meshes):
         text = get_text(root, 'ufs_mesh')
-        if text is not None:
-            path = f"./mesh[@id='{int(text)}']"
-            elem = root.find(path)
-            if elem is not None:
-                self.ufs_mesh = RegularMesh.from_xml_element(elem)
-        if meshes is not None and self.ufs_mesh is not None:
-            meshes[self.ufs_mesh.id] = self.ufs_mesh
+        if text is None:
+            return
+        mesh_id = int(text)
+        if mesh_id not in meshes:
+            raise ValueError(f'Could not locate mesh with ID "{mesh_id}"')
+        self.ufs_mesh = meshes[mesh_id]
 
     def _resonance_scattering_from_xml_element(self, root):
         elem = root.find('resonance_scattering')
@@ -1743,16 +1741,13 @@ class Settings:
 
     def _weight_windows_from_xml_element(self, root, meshes=None):
         for elem in root.findall('weight_windows'):
-            ww = WeightWindows.from_xml_element(elem, root)
+            ww = WeightWindows.from_xml_element(elem, meshes)
             self.weight_windows.append(ww)
 
         text = get_text(root, 'weight_windows_on')
         if text is not None:
             self.weight_windows_on = text in ('true', '1')
 
-        if meshes is not None and self.weight_windows:
-            meshes.update({ww.mesh.id: ww.mesh for ww in self.weight_windows})
-
     def _weight_window_checkpoints_from_xml_element(self, root):
         elem = root.find('weight_window_checkpoints')
         if elem is None:
diff --git a/openmc/stats/multivariate.py b/openmc/stats/multivariate.py
index 22c3d7ea8..3922d601a 100644
--- a/openmc/stats/multivariate.py
+++ b/openmc/stats/multivariate.py
@@ -744,7 +744,7 @@ class MeshSpatial(Spatial):
 
         # check if this mesh has been read in from another location already
         if mesh_id not in meshes:
-            raise RuntimeError(f'Could not locate mesh with ID "{mesh_id}"')
+            raise ValueError(f'Could not locate mesh with ID "{mesh_id}"')
 
         volume_normalized = elem.get("volume_normalized")
         volume_normalized = get_text(elem, 'volume_normalized').lower() == 'true'
diff --git a/openmc/weight_windows.py b/openmc/weight_windows.py
index f59364389..96f1db892 100644
--- a/openmc/weight_windows.py
+++ b/openmc/weight_windows.py
@@ -353,15 +353,15 @@ class WeightWindows(IDManagerMixin):
         return element
 
     @classmethod
-    def from_xml_element(cls, elem: ET.Element, root: ET.Element) -> WeightWindows:
+    def from_xml_element(cls, elem: ET.Element, meshes: Dict[int, MeshBase]) -> WeightWindows:
         """Generate weight window settings from an XML element
 
         Parameters
         ----------
         elem : lxml.etree._Element
             XML element
-        root : lxml.etree._Element
-            Root element for the file where meshes can be found
+        meshes : dict
+            Dictionary mapping IDs to mesh objects
 
         Returns
         -------
@@ -370,10 +370,9 @@ class WeightWindows(IDManagerMixin):
         """
         # Get mesh for weight windows
         mesh_id = int(get_text(elem, 'mesh'))
-        path = f"./mesh[@id='{mesh_id}']"
-        mesh_elem = root.find(path)
-        if mesh_elem is not None:
-            mesh = MeshBase.from_xml_element(mesh_elem)
+        if mesh_id not in meshes:
+            raise ValueError(f'Could not locate mesh with ID "{mesh_id}"')
+        mesh = meshes[mesh_id]
 
         # Read all other parameters
         lower_ww_bounds = [float(l) for l in get_text(elem, 'lower_ww_bounds').split()]