I'll start by saying that the equations you wrote make no sense. Not only did you mistype compounds, but you wrote the same incorrect reaction twice.
Your target reaction is
HCl + NaNO_2 -> HNO_2 + NaCl, " "DeltaH_"rxn"=?
The four reactions that you're supposed to use look like this:
color(blue)((1)) 2NaCl + H_2O -> 2HCl + Na_2O, " "DeltaH_1 = "+507 kJ"
color(blue)((2)) NO + NO_2 + Na_2O -> 2NaNO_2, " "DeltaH_2 = "-427 kJ"
color(blue)((3)) NO + NO_2 -> N_2O + O_2, " "DeltaH_3 = "-43 kJ"
color(blue)((4)) 2HNO_2 -> N_2O + O_2 + H_2O, " "DeltaH_4 = "+34 kJ"
Take a look at the four reactions you have. Notice that the target equation has sodium chloride on the products's side and hydrochloric acid on the reactants' side.
Reaction color(blue)((1)) has them the other way around. Not only that, but this reaction has 2 moles of each, instead of 1 mole of each. Don't worry about that right now, just flip reaction color(blue)((1)). This will become
color(blue)((1^')) 2HCl + Na_2O -> 2NaCl + H_2O,
DeltaH_1^' = (-DeltaH_1) = "-507.31 kJ" = "-507 kJ"
Now focus on sodium nitrite. Once again, the target reaction has it on the reactants' side, but reaction color(blue)((2)) has it on the products's side.
Flip reaction color(blue)((2)) to get
color(blue)((2^')) 2NaNO_2 -> NO + NO_2 + Na_2O
DeltaH_2^'= (-DeltaH_2) = "+ 427.14 kJ" = "+427 kJ"
Finally, focus on nitrous acid. You need it on the products' side, but have it on the reactants' side in reaction color(blue)((4)). Once again, flip reaction color(blue)((4)) to get
color(blue)((4^')) N_2O + O_2 + H_2O -> 2HNO_2
DeltaH_4^' = (-DeltaH_4) = "-34.25 kJ" = "-34 kJ"
You have everything that you need for the target reaction, so leave reaction color(blue)((3)) unchanged.
Add all these reactions and eliminate compounds that are on both sides of the general reaction to get
2HCl + cancel(Na_2O) + 2NaNO_2 + cancel(NO) + cancel(NO_2) + cancel(Na_2O) + cancel(O_2) + cancel(H_2O) = 2NaCl + cancel(H_2O) + cancel(NO) + cancel(NO_2) + cancel(Na_2O) + cancel(N_2O) + cancel(O_2) + 2HNO_2
This is reduced to the form
2HCl + 2NaNO_2 -> 2NaCl + 2HNO_2
DeltaH_"rxn" = DeltaH_1^' + DeltaH_2^' + DeltaH_3 + DeltaH_4^'
DeltaH_"rxn" = "-507" + 427 + ("-43") + ("-34 kJ")
DeltaH_"rxn" = "-157 kJ"
All you have to do now is divide this reaction by 2 to get the final form
HCl + NaNO_2 -> NaCl + HNO_2
DeltaH_"rxn" = ("-157 kJ"/2) = color(green)("-78.5 kJ")
