finches

Brambling and Siskin flocks are frequent sight in the hills throughout the region in winter. They are more likely within the city later in the season and can often be found in riverside trees, temple gardens or parks but occurrence and numbers vary depending on weather and food availability elsewhere. I would be confident of seeing both species in many mountainous areas but closer to home, say at Takaragaiike, the botanical gardens or along the Katsura River, it's a matter of pot-luck. Though once they've arrived in an area they're likely to stay as long as the food lasts, which usually measns all winter in their case.
Oriental Greenfinch is a very common species in the region, found in a variety of habitats from open arable land to forested hill tops. It's also common along Kyoto's rivers and in the parks, in fact hard to avoid might be a better way to describe its status. Winter flocks can be very large.
Common Redpoll is an irregular winter visitor to Kansai, it is sometimes found wintering on easily accessible mountains in the south of the region and must no doubt be overlooked in less accessible locations. While I've never heard of lowland birds further south than Lake Biwa it can occur at coastal locations on the Japan Sea.
Asian Rosy Finch is unusual in the region but is more regular on Ibukiyama and other high peaks on the periphery of Kansai. I'm told they occur in forested mountains in central Kyoto prefecture but these areas are largely inaccessible due to snow.
Wintering Long-tailed Rosefinches are far more widespread and in Kyoto can be found in scrubby riparian habitat and woodland edges around the city. It doesn't get into city parks but I'd expect to see it easily in suitable areas.Common Rosefinch definitely can't be expected in the region but odd ones are quite regular on off-shore islands particularly in spring.Pallas's Rosefinch is less regular than further north in Honshu. I've only seen them twice, in the mountains on the Kii Peninsular and to the north of Lake Biwa.
Pine Grosbeak is eruptive but rare even in the few years they do turn up and there might just be one or two singles throughout the region. The only flock I've heard of in Kyoto city was about 30 years ago.
Common Crossbill occurrence is slightly more frequent and numbers can be quite large in good years, the winter of 2013/14 has been the best I've known with flocks in many places throughout the region, with a Two-barred in a flock near Lake Biwa and another in nearby north east Shikoku.
Eurasian Bullfinch P.p.rosacea is fairly widespread, sometimes common, in winter but numbers do vary from year to year. Some authorities lump rosacea and griseiventris however when comparing the more distinctive males there are certainly morphological differences between birds I see in Hokkaido and Kansai. I've only once seen a bird with completely red underparts, on a headland in Shimane one February, unfortunately it wasn't actually a birding trip and I'm unsure which taxon it was. 
Hawfinch is a fairly common winter visitor in Kansai, singles or small parties are regular in city parks and wooded temple gardens or less regularly in trees lining rivers.
Japanese Grosbeaks breed in forested hills Kansai and are common in wooded city parks such as the botanical gardens and the Imperial Palace Park in Kyoto. Chinese is very scarce in Kansai, I don't hear of them every year. I once found a pair wintering in a small Kyoto shrine garden, but they are more often found in Osaka parks.

On Hegura flocks of Bramblings can start to arrive from early October, this one of many birds on the 13th.

A Brambling from a flock which spent the winter in a small area of trees the Katsura River (Kyoto), this shot 30 December.

On 17 April there were still some lingering birds in the same trees.

There can be huge flocks in Lakeside trees around Lake Biwa and elsewhere.

Birds in a small clump of trees at the mouth of the Yamato River in Osaka city.

Juvenile Siskin on Hegura, 14 October 2013.

Male Siskin, Hokkaido 31 December 2012.

Two members of a flock wintering in riverside trees on the Katsura River (Kyoto city), 17 January 2011.

An exceptionally dark bird on Mishima (Yamaguchi), 29 April 2011. Not only is the mantle black but the yellows and greens are much duller than usual.

A party of Common Redpolls spent weeks in a tiny group of small trees next to Izumo Airport (Shimane), 4 January 2009.

Common Redpoll, Izumo Airport. I believe there had been Arctic Redpoll with the birds when they first arrived but they didn't stay as long.

Long-tailed Rosefinch is common along the Katsura River (Kyoto city) in winter.

Female Long-tailed Rosefinch along the Katsura River, Kyoto city.

Unlike most migrant Common Rosefinches I see, this was with a flock of Eurasian Tree Sparrows in the port on Mishima (Yamaguchi), 3 May 2011.

Pine Grosbeaks in Shari port, Hokkaido, 30 December 2012.

One of a large number of birds at Woody Pal, Shiga, 23 November 2013.

Male griseiventris Bullfinch at Abashiri, Hokkaido, 31 December 2012. Clean grey breast and belly extending onto including lower throat. Clearly defined red area contrasts sharply with underparts. 

Male rosacea Bulfinch on Mt.Katsuragi (Nara), 15 February 2009. Unlike griseiventris it has a distinct reddish cast to breast and belly, the more extensive red on face and throat is less sharply defined.

Female rosacea Bullfinch with brownish cast to underparts. Lake Biwa (Shiga), 26 January 2009.

Members of a small group of very eye-catching Bullfinches at Woody Pal (Shiga), 2 December 2012. There was a large influx of Bullfinches that winter and this group differed from the usual Kansai birds in having uniformly grey underparts without any hint of reddish or brownish. Though they are slightly paler bordering the black cap, perhaps all female griseiventris?

Hawfinch in the Kyoto botanical gardens, 3 February 2013.

Hawfinch on Mishima (Yamaguchi), 29 April 2011.

Chinese Grosbeak Senri Park (Osaka), 14 January 2007. To my knowledge birds came to this park in two consecutive winters and also to another park in the north of the city for two further winters.

Japanese Grosbeaks in Kyoto botanical gardens, 7 March 2012. The gardens are a great place to see the birds on the ground as they get accustomed to people slowly wandering round.