French bean Cobra and Sweet Corn Lark are just two of the reliable and delicious crops you could sow now directly into polytunnel soil.
Sow in a heated propagator, in a warm place, or directly in tunnel soil when it's warm enough - for polytunnel or greenhouse cropping, or for planting outside under cloches or fleece at the end of May you can now sow:
French beans, runner beans, sweet corn, courgettes, edamame (soy) beans, chick peas, cucamelons, gherkins, pumpkins, summer squashes, marrows and melons. You can still sow cucumbers and tomatoes for late tunnel/greenhouse crops. Also 'soft' herbs such as basil, coriander, dill, Greek oregano (best for flavour), lovage, mints, parsley (giant flat-leaf Italian best flavour) Perilla (or Japanese beefsteak plant) and fennel, Alpine strawberries (Reugen is a good large-fruited variety).
Also Florence fennel and half-hardy single flowers such as Tagetes, single French marigolds, nasturtiums etc. for bees and butterflies - and to attract other beneficial insects like hoverflies etc. to help with pest control and pollination, both under cover and out in the garden.
It's really important to shade propagators and young seedlings from strong sun at all times now to stop seedlings from cooking! - You can also switch off propagators during the day to save energy - even if shaded, on sunny days they will be plenty warm enough - but do make sure you remember to turn them on again well before it gets chilly in the evening, in case of unexpected frost..
Sow in modules if the weather is still too cold, in a seedbed for transplanting, or in the ground where they are to crop - if the weather and your ground conditions are suitable: Asparagus, globe artichokes, beetroot, broad beans, carrots, all varieties of peas, savoy and other autumn/winter cabbages, all varieties of sprouting broccoli including calabrese and purple sprouting, cauliflowers, salad onions, Hamburg parsley, landcress, lettuces, perilla, orache, chicory, kohl-rabi, kales (those for cropping overwinter outside from the middle of May onwards), parsnips (early May) radishes, rocket, salsify, Swiss chards, spinach, turnips and swedes, summer purslane, lamb's lettuce, salad mixes and perennial hardy herbs including sorrel. Asparagus peas, Cardoons, Good King Henry and New Zealand spinach can be sown outside under cloches now, and also from the middle of May, if the soil is warm enough, sweet corn, French and runner beans. Marrows, courgettes, pumpkins, ridge cucumbers and squashes can all be sown outside under cloches at the end of May, in warm areas. Also sow some single annual flowers such as Sweet Alyssum (perennial in polytunnels, sowing itself freely), limnanthes (poached egg flower), cosmos, calendula, Californian poppies, Convulvulus tricolour, nasturtiums, phacelia, sunflowers etc. All of these will attract beneficial insects like hoverflies, lacewings and ladybirds which will help with pest control, and also attract bees which help with crop pollination. Sow fast-growing green manures like buckwheat, mustard (a brassica so make sure it fits into your rotations), lupins & red clover (legumes) and phacelia, to improve the soil by adding humus, to encourage beneficial microbes, 'lock-up' carbon and feed worms (after digging in), on any empty patches of ground that won't be used for 6 weeks or more.
Unless you have had ground covered with cloches or polythene so that soil is dry and really warm - and the space is relatively slug-free - then it's always much safer to sow in modules undercover in a greenhouse, polytunnel or cold frame to be sure of guaranteed results and to save wasting expensive seed. Also don't make the mistake of sowing too deep. This is one of the main reasons for seed not germinating. Never sow seed at more than twice it's own depth - and some seeds like celery and alpine strawberries actually prefer to be sown on the surface of compost, or only very lightly covered with something like vermiculite or sharp sand which allows light to filter in.
The soil in some areas is still cold after the recent frosts. A late frost could destroy newly emerged seedlings of tender crops like French beans even under cloches. Seed is expensive and you can't afford to waste it. You can't afford to waste time now either, by possibly losing any sowings made at this time of year - especially now, as with the increase in people growing their own food due to the COVID19 pandemic and supply chain issues due to the war in Ukraine - some seed has become almost impossible to get!. any important staple winter and storage crops need to be sown this month - and if they fail it may well be too late to sow them again, even if you can get seed. Although the sun is strong now and sunny days are warm, there can still be seriously damaging night frosts
In the tunnel you can plant tender veg like sweet potato 'slips' in pots this monthvery - or in the ground if it's reliably warm enough, but I always prefer to get mine growing really well in pots first, as it gives them a better start. They need a long season as they don't bulk up until late, frost tender and hate cold, wet ground. You can also plant Oca and Mashua tubers in pots now - again to plant out later, at the end of May or early June - or to plant directly into tunnel soil. The small growing tubers of Yacon can also be planted now in the tunnel or in pots to plant outside later. They are just starting into growth now.