diff --git a/nominatim/api/logging.py b/nominatim/api/logging.py
index a4da357d..6c8b1b38 100644
--- a/nominatim/api/logging.py
+++ b/nominatim/api/logging.py
@@ -178,7 +178,7 @@ class HTMLLogger(BaseLogger):
             self._write(f"rank={res.rank_address}, ")
             self._write(f"osm={format_osm(res.osm_object)}, ")
             self._write(f'cc={res.country_code}, ')
-            self._write(f'importance={res.importance or -1:.5f})</dd>')
+            self._write(f'importance={res.importance or float("nan"):.5f})</dd>')
             total += 1
         self._write(f'</dl><b>TOTAL:</b> {total}</p>')
 
diff --git a/nominatim/api/search/db_search_builder.py b/nominatim/api/search/db_search_builder.py
index b6ba211c..9ff8c03c 100644
--- a/nominatim/api/search/db_search_builder.py
+++ b/nominatim/api/search/db_search_builder.py
@@ -141,12 +141,14 @@ class SearchBuilder:
             yield dbs.CountrySearch(sdata)
 
         if sdata.postcodes and (is_category or self.configured_for_postcode):
+            penalty = 0.0 if sdata.countries else 0.1
             if address:
                 sdata.lookups = [dbf.FieldLookup('nameaddress_vector',
                                                  [t.token for r in address
                                                   for t in self.query.get_partials_list(r)],
                                                  'restrict')]
-            yield dbs.PostcodeSearch(0.4, sdata)
+                penalty += 0.2
+            yield dbs.PostcodeSearch(penalty, sdata)
 
 
     def build_housenumber_search(self, sdata: dbf.SearchData, hnrs: List[Token],
diff --git a/nominatim/api/search/db_searches.py b/nominatim/api/search/db_searches.py
index db35f726..ba4a3995 100644
--- a/nominatim/api/search/db_searches.py
+++ b/nominatim/api/search/db_searches.py
@@ -562,6 +562,8 @@ class PlaceSearch(AbstractSearch):
             sql = sql.where(tsearch.c.country_code.in_(self.countries.values))
 
         if self.postcodes:
+            # if a postcode is given, don't search for state or country level objects
+            sql = sql.where(tsearch.c.address_rank > 9)
             tpc = conn.t.postcode
             if self.expected_count > 1000:
                 # Many results expected. Restrict by postcode.
diff --git a/nominatim/api/search/token_assignment.py b/nominatim/api/search/token_assignment.py
index 747fea6c..11da2359 100644
--- a/nominatim/api/search/token_assignment.py
+++ b/nominatim/api/search/token_assignment.py
@@ -270,7 +270,12 @@ class _TokenSequence:
             if (base.postcode.start == 0 and self.direction != -1)\
                or (base.postcode.end == query.num_token_slots() and self.direction != 1):
                 log().comment('postcode search')
-                yield dataclasses.replace(base, penalty=self.penalty)
+                # <address>,<postcode> should give preference to address search
+                if base.postcode.start == 0:
+                    penalty = self.penalty
+                else:
+                    penalty = self.penalty + 0.1
+                yield dataclasses.replace(base, penalty=penalty)
 
         # Postcode or country-only search
         if not base.address:
@@ -278,6 +283,9 @@ class _TokenSequence:
                 log().comment('postcode/country search')
                 yield dataclasses.replace(base, penalty=self.penalty)
         else:
+            # <postcode>,<address> should give preference to postcode search
+            if base.postcode and base.postcode.start == 0:
+                self.penalty += 0.1
             # Use entire first word as name
             if self.direction != -1:
                 log().comment('first word = name')
diff --git a/test/python/api/search/test_token_assignment.py b/test/python/api/search/test_token_assignment.py
index f78d5430..dc123403 100644
--- a/test/python/api/search/test_token_assignment.py
+++ b/test/python/api/search/test_token_assignment.py
@@ -253,7 +253,7 @@ def test_postcode_with_designation():
                    (BreakType.PHRASE, PhraseType.NONE, [(2, TokenType.PARTIAL)]))
 
     check_assignments(yield_token_assignments(q),
-                      TokenAssignment(name=TokenRange(1, 2),
+                      TokenAssignment(penalty=0.1, name=TokenRange(1, 2),
                                       postcode=TokenRange(0, 1)),
                       TokenAssignment(postcode=TokenRange(0, 1),
                                       address=[TokenRange(1, 2)]))
@@ -266,7 +266,7 @@ def test_postcode_with_designation_backwards():
     check_assignments(yield_token_assignments(q),
                       TokenAssignment(name=TokenRange(0, 1),
                                       postcode=TokenRange(1, 2)),
-                      TokenAssignment(postcode=TokenRange(1, 2),
+                      TokenAssignment(penalty=0.1, postcode=TokenRange(1, 2),
                                       address=[TokenRange(0, 1)]))