Only parameterless constructors and initializers are supported in LINQ to Entities

without more info on 'Payments' this doesn't help much, but assuming you want to create a Payments object and set some of its properties based on column values:

var naleznosci = (from nalTmp in db.Naleznosci
                              where nalTmp.idDziecko == idDziec
                              select new Payments
                              {
                                  Imie = nalTmp.Dziecko.Imie,
                                  Nazwisko = nalTmp.Dziecko.Nazwisko,
                                  Nazwa= nalTmp.Miesiace.Nazwa,
                                  Kwota = nalTmp.Kwota,
                                  NazwaRodzajuOplaty = nalTmp.RodzajeOplat.NazwaRodzajuOplaty,
                                  NazwaTypuOplaty = nalTmp.RodzajeOplat.TypyOplat.NazwaTypuOplaty,
                                  DataRozliczenia = nalTmp.DataRozliczenia,
                                  TerminPlatnosci = nalTmp.TerminPlatnosci,
                              }).ToList();

If you still want to use your constructor for initialization and not properties (sometimes this behaviour is desired for initialization purposes), enumerate the query by calling ToList() or ToArray(), and then use Select(…). Thus it will use LINQ to Collections and that limitation of not being able to call constructor with parameters in Select(…) will vanish.

So your code should look something like this:

var naleznosci = db.Naleznosci
                          .Where(nalTmp => nalTmp.idDziecko == idDziec)
                          .ToList() // Here comes transfer to LINQ to Collections.
                          .Select(nalImp => new Payments
                              (
                                  nalTmp.Dziecko.Imie,
                                  nalTmp.Dziecko.Nazwisko,
                                  nalTmp.Miesiace.Nazwa,
                                  nalTmp.Kwota,
                                  nalTmp.RodzajeOplat.NazwaRodzajuOplaty,
                                  nalTmp.RodzajeOplat.TypyOplat.NazwaTypuOplaty,
                                  nalTmp.DataRozliczenia,
                                  nalTmp.TerminPlatnosci
                              ))
                          .ToList();

Having just encountered this error myself, I thought I would add that if the Payment type is a struct, you would also encounter the same error because struct types do not support parameterless constructors.

In that event, converting Payment to a class and using the object initializer syntax will resolve the issue.