There are a lot of (more or less obscure) words out there that mean look at angrily or look sideways/askance at, but I haven't been able to identify any with the exact meaning look sadly at.
Close contenders (all definitions from the OED online, but quotations found in Google Books, rather than the OED's attestations):
gloom, v. 1
- intr. To look sullen or displeased; to frown, scowl, lower; also to gloom on or to gloom at (a person). In recent use also (through influence of gloomy adj.): To look dismal or dejected,
to wear an air of sombre melancholy; to be gloomy.
Gloom is a fairly familiar term, at least as a noun, with clearly melancholic connotations. However, the usages I found for gloom at were all on the angry side of sad, and it doesn't have any implication that the gloomer is subordinate to the gloomee.
- c. To adopt a languid look, expression, or pose, as an indication of sorrowful or tender emotion. Now arch. and rare. In quot. 1714
trans.: to bestow upon with a languid look.
For a nearly contemporary example:
"Oh what d'you think? she languished at him in this quivering
schoolchild way the way they do, just pathetic." (William Spackman,
An Armful of Warm Girl, 1981.)
Languish has some of the connotation of an imbalance if power, but I think that in the languish at form it is usually the tender emotions being evoked, rather than sorrowful.
- Thesaurus »
1.a. intr. Of persons, their eyes, countenances, etc.: To frown, scowl; to look angry or sullen. †Also, to be depressed or mournful.
Const. at, on, upon; rarely in indirect passive.
Unfortunately, again most of the Examples I could find come much closer to glaring or scowling than what I think you're after, but this quotation does have some of the connotations I think you want:
Gunpat's house, with its sightless windows, its towers, its endless
terraces, its unnecessary ornamentation, its massive porch, lowered at
him like a forsaken wedding cake. (Brian Aldiss, Galaxies Like
Grains of Sand, 2014)
peep, v. 1
II. Senses relating to looking.
3. intr. To look through a narrow aperture, as through half-shut eyelids or through a crevice, chink, or small opening into a larger
space; (hence) to look quickly or furtively from a vantage point; to
steal a glance. Also in fig. context.
Given the sense of furtiveness and, I think, the association with baby peeps, this one has the strongest sense of the one looking being smaller in some way than the one being looked at, but unfortunately it does not have anything to do with sadness.
If course, you might productively combine any of these with an adverb to sharpen your meaning, if your main concern is avoiding the word look rather than having a unitary verb. Dolefully send especially apropos.
One other possibility would be to coin your own phrase; if you use the construction verbed/adjectived up at him I think you will get across the idea of looking and the power differential. So
The beggar yearned up at the landowner.
The beggar woebegoned up at the landowner.
And so forth.