What symbol expresses “less than approximately”?

In Unicode, the symbol $\lessapprox$ is named “LESS-THAN OR APPROXIMATE”, which describes pretty much the meaning you are after. It also is built from the symbols $<$ (less than) and $\approx$ (approximately equal) in the very same way as $\leqq$ (less or equal) is built from $<$ (less) and $=$ (equal).

Given that commonly for "less than equal" a single line is used instead of a double line (i.e. $\leq$ instead of $\leqq$), using $\lesssim$ instead of $\lessapprox$ seems justified as well.

Indeed, in the comments to your question, Akiva Weinberger has confirmed that the latter symbols is actually used with that meaning in "NSA contexts" (which probably means non-standard analysis).

So in summary, I'd say for both symbols there's strong evidence that their use with that meaning is appropriate.

Another option might be to write $a<b+\epsilon$. Commonly $\epsilon$ is understood as a small positive quantity, so I'd argue that you are not really introducing a new variable there.

'I want to state that $a$ is lesser than some value which is approximately $c$.'

Then write $$a<b\approx c.$$