How to set up an ICE table for adding a weak acid to a strong acid
ICE tables tabulate amounts or concentrations pertaining to an equilibrium reaction: at the initial stage (I), how they change (C), and what they are at the end or at equilibrium (E). They are an alternative to solving multiple simultaneous equations (equilibrium constant expression, mass balance), instead using a single parameter (related to the extent of reaction $\xi$).
For the problem at hand, there is one reaction that goes to completion (strong acid) and one that attains equilibrium (weak acid, both protonated and deprotonated species present at equilibrium).
The ICE table needs to have species from the second reaction only. A good estimate of the hydrogen ion concentration is the concentration of hydrochloric acid (it dissociates completely, and we are ignoring the autodissociation of water for now).
$\begin{array}{|c|c|c|c|} \hline
\ &amp; [\ce{HP}] &amp; [\ce{H+}] &amp; [\ce{P-}]\\ \hline
\text{I} &amp; 0.1 &amp; 0.02 &amp; 0 \\
\text{C} &amp; -x &amp; +x &amp; +x \\
\text{E} &amp; 0.1-x &amp; 0.02 +x &amp; x\\ \hline \end{array}\\\\$
If you do the math, you will find that very little of the propanoic acid will dissociate, as the $\mathrm{p}K_\mathrm{a}$ is far from the initial pH.