Let z be a complex number such that

Since, $|z|+z=3+i$

Let $z=a+i b$, then

$|z|+z=3+i \Rightarrow \sqrt{a^{2}+b^{2}}+a+i b=3+i$

Compare real and imaginary coefficients on both sides

$b=1, \sqrt{a^{2}+b^{2}}+a=3$

$\sqrt{a^{2}+1}=3-a$

$a^{2}+1=a^{2}+9-6 a$

$6 a=8$

$a=\frac{4}{3}$

Then,

$|z|=\sqrt{\left(\frac{4}{3}\right)^{2}+1}=\sqrt{\frac{16}{9}+1}=\frac{5}{3}$