Table[Total[DeleteDuplicates[{p[i1], p[i2]}]], {i1, 2}, {i2, 2}]

Table @@ {Total[DeleteDuplicates[{p[i1], p[i2]}]], {i1, 2}, {i2, 2}}

t[n_, m_] := Table @@ {Total[DeleteDuplicates[Table[p[i[j]], {j, m}]]], Sequence @@ Table[{i[j], n}, {j, m}]}

t[n_, m_] := Table[Total[DeleteDuplicates[Table[p[i[j]], {j, m}]]], Sequence @@ Table[{i[j], n}, {j, m}]]

t[4, 3]

n = 5
ps = Array[p[#] &, n]
Total /@ DeleteDuplicates /@ Tuples[ps, n]