Use the GroupBy.transform with sum for new column filled by aggregated values, compare for greater by Series.gt and filer by boolean indexing:
df[df.groupby('Neighbourhood')['Phone connection'].transform('sum').gt(10)]
Sample: with greater like 3:
print (df)
House Neighbourhood Phone connection
0 #1 A 1
1 #2 B 0
2 #3 B 1
3 #4 A 1
4 #5 C 1
5 #6 C 0
6 #7 A 0
7 #8 B 1
8 #9 B 1
9 #10 A 1
10 #11 A 1
11 #12 B 0
12 #13 B 1
13 #14 A 1
14 #15 C 1
15 #16 C 1
df = df[df.groupby('Neighbourhood')['Phone connection'].transform('sum').gt(3)]
print (df)
House Neighbourhood Phone connection
0 #1 A 1
1 #2 B 0
2 #3 B 1
3 #4 A 1
6 #7 A 0
7 #8 B 1
8 #9 B 1
9 #10 A 1
10 #11 A 1
11 #12 B 0
12 #13 B 1
13 #14 A 1
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