For integer-valued polynomials $f_1(x),f_2(y),f_3(z)$, if any $n𝟄\mathbb N=\{0,1,2,.\}$ can be written as $f_1(x)+f_2(y)+f_3(z)$ with $x,y,z𝟄\mathbb N$ then we say that $f_1(x)+f_2(y)+f_3(z)$ is universal (over $\mathbb N$). In this paper we find all candidates of universal sums of the following three types: $ax^2+by^2+f(z),\ aT_x+bT_y+f(z),\ aT_x+by^2+f(z),$ where $T_x$ denotes $x(x+1)/2$ and $f(z)$ has the form $c\binom z2+dz$ with $c,d𝟄\{1,2,3,.\}$ and $d∤ c$. We also show that some of the candidates (including $x^2+y^2+z(z+3)/2$, $T_x+y^2+z(z+2k)$ for $k=1,2,3$, and $T_x+y^2+z(z+2k+1)/2$ for $k=1,.,7$) are indeed universal sums. ; Comment: 21 pages. Add new results to Theorem 1.4