Large aperture - small depth of field, so you need to get the focus spot
on and it just won't work with deep subjects.
Fast ISO - more noise, so more post-processing required.
Slow shutter - motion blur if the camera or subject moves even slightly.

You have to choose the lesser evil based on your subject and the effect
you want to achieve - motion blur is commonly used as a creative effect
in night shots, if you're going for a twinkly-lights effect then extra
noise may not matter, if you're shooting an illuminated statue/building
etc. then small DOF may be good ...

There are some very good noise reduction programs available, also available
as plug-ins for photoshop and others.

Its not a perfect solution, but this will allow you to kick up the gain
setting (ISO) on your camera. When you increase the ISO, you also bring up
the noise floor, hence the visible noise.

I like NeatImage myself.

Vice versa..."stopped down" = small aperture (in size, big in number).  If
you increase the ISO, the "film" is faster so the lens can be stopped down
more.

I'm not sure exactly what being not as clear as you would like means,
but if it means that they aren't as sharp as you would like, then you
have two problems.  One is that virtually all digital images need some
degree of sharpening because of the effects of in-camera processing
and it is a routine step in most workflows.

The second problem you probably face is due to using the largest
aperture your lens provides.  Almost no lenses are as sharp wide open,
or stopped down to the smallest available aperture, as they are in the
middle range.  A decent rule of thumb is that the apertures of f8
through f16 will give the best resolution a lens is capable of.
Shooting within the sharpest range of the lens plus post capture
sharpening will give you the best results.

If you are using a tripod the length of exposure shouldn't be a big
deal for a cityscape.  It's true that a long exposure, say over 2
seconds (depends on the camera), will have more noise than a shorter
exposures, most decent cameras have a way to reduce that at the time
of capture.  My Canon takes two images, one blank, and uses that to
reduce noise.

If you increase ISO to get either a smaller aperture or a shorter
shutter speed, the noise introduced can be worse than what you get by
using a longer exposure.  Again, it depends on the camera.

Vance

FWIW, my tactics for low light (or time exposure) shots;

I keep the ISO at 100 or 200 so as to avoid additional noise - UNLESS I
want that slightly grainy look. Which I often do, especially if I think
I might want to render the photo B&W.

I use a solid tripod. I use an electronic shutter release, but first
flip the mirror (DSLR) so as to avoid the additional shake mirror
opening causes.

I almos never have the aperture wide open - as another reply pointed
out, in low light that can cause finding focal point too difficult. But
I sometimes close it wat down; to increase focal length, but also to
force a longer exposure (such as blurring moving water, or getting
light streaks from stars).

I allow the exposure to be long enough to get the desired effect
(moving water blurred, low-light shots becoming 'bright as day', night
shots with specific lit points, etc).

I do not find the need for additional noise reduction sotware if I get
a clean picture in the first instance.